Disclaimer: I would love the experts of the matchup to prove me wrong. What I am going to say is not going to be perfect, I've only fought a few Bowsers in my day. This is simply my knowledge of Mewtwo being put in perspective of what I know fighting a Bowser.
Rather than talking about everything in a bunched up paragraph, I'm going to break it down to a comparison between some very relevant mechanics between the characters, such as movement speed, recovery, and projectile game. I will end the post with my opinion on the matchup in conclusion. This is simply to help contribute to the forum post and not to be an end all post, take this (practically subjective) post with a grain of salt.
EDIT: I compare moves here. I am not comparing a nair versus another nair literally attacking each other, I'm comparing the practical use of a nair versus any move or situation in the game, VERSUS the same thing for the other character. Please don't confuse the two.
Mechanic/Move/Etc - Character in favour
Movement Speed - Mewtwo
Mewtwo is very fast with the new buff, and it makes a few interesting adjustments. The ability to react and take action on an opponents mistake is much more possible with this speed, and moves like up smash on platforms can be pulled off as a punish more often on opponent who are silly enough to not tech on said platform on the other end of the stage. Movement speed in general brings more situational options, but it can also catch your opponent off guard. Bowser is honestly pretty quick, but Mewtwo is faster and makes use of it much better.
Recovery - Mewtwo
An outstanding jump, a fantastic side b for flying above annoying edge-guarding smash attacks and other shenanigans, plus the ability to go under some stages, then a really smooth up b that also can dodge certain situations out of your favour. Bowser has a terrible mid-air jump, and his up b just isnt quite good enough vertically.
Weight - Bowser
Mewtwo unfortunately has a weight problem -- He needs more McDonalds in his diet apparently. Though he does have the ability to get out of certain combos from some opponents, Bowser doesnt exactly have to worry about his combo potential in order to hurt Mewtwo. Mewtwo dies pretty early and his floaty exterior makes him easy to prepare for, even when fast falling back to the stage from high up. Bowser doesnt die for a while and his weight only really affects his recovery, since his move speed is actually decent for a heavy character.
Ability to Punish - Tied
Up for debate. I am aware Bowser gains a lot of his potential from punish, but I know very little about such info on Bowser so its not fair for me to speak.
Spacing - Mewtwo
Bowser isnt equipped well for range, but he does have a few options such as neutral b and fair. and that's why he can easily be considered combo food over some of the other heavy-weights. However, Mewtwo has a great group of moves to pull off to keep the opponent at bay, and even has some good camping moves. Moves like Shadow Ball, Shadow Claw and Confusion are fantastic for making sure your opponent keeps their distance.
Now its time to talk about individual moves.
Jab - Tied
Bowser and Mewtwo use their jabs in specific ways that make it very different to compare. Bowser can use it to stop some projectiles, and get out of some painful situations, while Mewtwo uses it to set up for throws, command grabs and again, getting out of painful situations. I think this is up for debate, to be honest. As far as I know, Bowser uses jab more often than Mewtwo, but I know Mewtwo jab has quite a few uses for set ups, so Im gunna leave it as a tie for now.
Down Tilt - Tied
Again, both characters use this move for several reasons and both of them use them a lot. Mewtwo uses it for crazy good range, throwing someone into the air for strings, and as a very fast poke. Bowser can use it as an extremely useful edge-guarding tool, and as a fast and strong poke. I'll leave this up for debate, personally I think Mewtwo is in favour but that could potentially be bias.
Side Tilt - Bowser
Bowser is in favour simply because he gets more percent. Mewtwo does get more range.
Up Tilt - Mewtwo
Bowser's Up Tilt is honestly pretty bad when you look at frame data and hitboxes vs Mewtwo's. UTilt starts at frame 11 for Bowser, frame 8 for Mewtwo. Damage is 9 for Bowser, and a max of 6 for Mewtwo, which would be better if the range was larger for Bowser. The FAF is 41 for Bowser, and 30 for Mewtwo. Hands down, Mewtwo has it better, and will most likely use it in more situations.
Dash - Mewtwo
You might notice a growing trend here. Both characters have similar damage output and frame data for each of these moves, and both moves go a pretty good distance too. But that FAF on Bowser... 56 frames. Almost a whole second before another move is ready. That is slightly more than triple the active frames on said move. Mewtwo is not nearly as detrimental, at 39 frames. Plus the angle the dash makes an opponent fly is debatably better on Mewtwo's side than Bowser.
Neutral Special - Mewtwo
Mewtwo can camp Bowser since Bowser relies on a lot of your mistakes, allowing Mewtwo to charge his shadow ball more often than most other characters in the entire game. Bowser may have flames, but its clear that Mewtwo is the winner here.... Not to mention Bowser cant stop the larger shadow balls from coming at him and his jump is too bad to jump over it unless hes real quick.
Side Special - Bowser
This is interesting. Both characters have a command grab, but Bowser will get much more out of it than Mewtwo. While Mewtwo can use it as a recovery method, and a way to get out of sticky situations, Bowser can use it to kill, which Mewtwo really cant do much of with his command grab.
Down Special - Bowser
Let's be honest, unless you're playing doubles, you probably won't use down special with Mewtwo almost ever. Maybe once per game at most. Bowser does have a lot of end lag, sure, but he can cancel it on ledges and can use it as a surprise attack on opponents to kill.
Up Special - Mewtwo
Although Bowser can up-b out of shield, Mewtwo can recover higher and he can use it to get out of edge-guarding shenanigans, which helps add a new layer of freshness to his options.
Nair - Mewtwo
Bowser has a nair? No but seriously, it isnt the best nair in the world, it doesnt really have a bunch of opportunity there. Mewtwo on the other hand, can use it in a variety of ways. Send people into the air, send people right in front of you on your way down, etc. Hands down, Mewtwo has a more practical nair.
Fair - Tied
Both characters really have a good fair, and as far to my knowledge put them to similar use. Mewtwo does have a nice use of being able to combo the move into itself early percents, though. If someone could tell me the kill% of Mewtwo's fair then that would be great, I honestly have no idea lmfao. I do know that Bowser has a pretty good fair though, hence why I dont want to right out and say its not as good. Up for debate, guys.
Dair - Mewtwo
Both characters make some use out of their down aerials, but Mewtwo won't suicide from the move, nor will he be suffering huge endlag after said move. However, one thing to note is the hitbox for Dair on Mewtwo isnt really that long, so that kinda sucks. But it's still a good attack overall, and honestly even if it doesnt spike, it has a pretty powerful upwards knockback hitbox regardless. Bowser also lacks a lot of horizontal control while using his Dair.
Uair - Tied
Another one that's hard to comment on. Believe it or not, Bowser's up air is not exactly strong, not to mention it lacks some range. It still would've been better than Mewtwo's up air, if it wasnt for the latest patch making his up air relevant by a pretty big degree. I'm pretty sure you can even use it as a kill move at higher percents now, usually when you catch people on platforms. To be fair, I use it more in competitive play than before now, so it could be in Mewtwos favor, but I'm gunna leave this one up to debate as well.
Bair - Bowser
Mewtwo has a pretty good back air actually, but its not very fast and its not too powerful even at high percents, since the base of his tail is basically the only relevant kill option. Bowser has some good range on his bair, and not only that, it kills pretty early, being his strongest aerial. It comes out at frame 9, which is not bad either, and I guess this is a good time to say his moves have pretty bad FAF in the air but thanfully the autocancel frames can be helpful, letting you land and shield 14 frames earlier than waiting for your FAF, in bair's case.
F Smash - Bowser
Both of these characters suffer from pretty laggy smash attacks, but Bowser gets much more rewards for his risky business in this category. I should mention Mewtwo has a nicer F Smash now than pre-patch, since it barely hit anyone half the time before, with the bigger hitbox its actually not bad. Bowser on the other hand, does not care. If he can catch you off guard, he will destroy you. Well, at early percents compared to other characters anyways.
D Smash - Mewtwo
Both characters have a pretty good dair, but Mewtwo can take advantage of it more often. Bowser basically uses up b out of shield instead of this move (as far as I know, if I am wrong about this please correct me), and Mewtwo doesnt really have a great replacement for this move other than maybe F Smash which doesnt have the same KB. Mewtwo down smash is honestly super strong, and isnt all that slow, plus it will hit the smaller characters, which the almighty up smash will not. So, this is a great move in terms of usage and power.
Up Smash - Mewtwo
I think it is undeniable when I say that Mewtwo has one of the best up smashes in the entire game. Not only is it really good for catching people who jump off the ledge, not only is it good for catching people on platforms in some stages, not only does it troll all the tall characters incredibly hard, it also kills pretty early considering most characters nowadays can't really offer the same level of vertical KB on their smash attacks. Plus, it's fun to catch Peach players who float cancel just barely overtop of me and I up smash them to oblivion rather than up tilting or something.
Sigh, yes I should probably talk about Bowser up smash too. It isnt exactly that bad, but it isnt very good either. It has a bit of start-up, in which the first hitbox frame is 16. At the very least, the move has awesome invincibility frames, from 14-27, nearly a quarter of a second. However, its really slow to finish, and the payoff is almost not worth it, since it kills at around the same time as Bair and with some more percent racked up, you could do better with dtilt or even dair. Even down b might be more worth it for you. Bottom line, Mewtwo has a more useful up smash for a few good reasons.
Throws - Mewtwo
Honestly, both characters have some pretty nice throws, with Bowser having 3 throws that could help seal the deal near the edge of the stage at higher percents, and Mewtwo having a variety of throws for every situation. But that is what makes the difference here: While Bowser does have some great fun with his up throw, Mewtwo has an outstanding down throw which can chain into up smash on some characters, a fair that can rack up damage pretty well, a back throw that is surprisingly useful near the edge of the stage, and the once terrifying up throw which is still pretty good today, being able to finish medium weights off at about 125% give or take. Mewtwo has a great variety of throws which makes his grab more relevant.
As such, I'm going to go on a whim and say Mewtwo has a better grab since both characters have nearly identical grabs and Mewtwo has the throws that really make sense of it all.
Final Thoughts:
This is where things can get very biased, but please bear with me. Bowser does seem to have some fantastic attributes in certain categories, presumably his KB on most of his moves and his ability to attack out of a lot of situations, plus having hands of steel apparently. However, Mewtwo is a bit more versatile in other categories and that can make or break the difference for a match between the two characters. Bowser needs range, and Mewtwo is making sure Bowser never gets close enough to him to get destroyed at early percent due to the light weight. However, Bowser can devour a lot of characters simply from the use of rage, and this is a huge factor I glossed over most of my study. Mewtwo has some pretty good moves to kill Bowser with, but Bowser could use his damn good command grab and his bair for some early kill shenanigans once Bowser has hit the 120+% mark.
Even though rage plays a role in this right, and Bowser has some pretty good tools to keep people off-stage such as fair and dtilt, I still think that Mewtwo has the potential to beat Bowser more often than not. His moves are punishable, sure, but Bowser lacks the speed to do this, basically leaving him to spam dash as a punish move, which isnt really much of anything... Or up b out of shield, which is actually really good against Mewtwo, as well as other characters. Plus, Bowser's flame breath can be a hilariously stupid gimp on Mewtwo if he decides to recover low for some reason. At the very least, Mewtwo has some crazy good mobility and recovery compared to Bowser and this makes some difference with the opportunity to space out Bowser and camp the crap out of him with Shadow Ball.
I think overall, the match-up is definitely something like 55:45 in favor of Mewtwo. If Mewtwo gets buffed in some places where it helps, or if Bowsers gets a really unfortunate nerf to some of his most useful moves like fair, dtilt, jab or his command grab, I would go as far to say Mewtwo would be 60:40. I think Bowser should not be the choice to fight Mewtwo if you felt the need to switch, there are a handful of characters that would have a better match-up than him.
