Honestly, I think Zard is slightly better. What matchups does Bowser win that Zard doesn't?
As a person who has played both Charizard and Bowser extensively, I believe that Sonic and Yoshi are the only two characters against whom Bowser
may do better than Charizard, as Bowser's amazing Whirling Fortress out of shield is incredibly useful for dealing with the pressure they apply. But aside from that, I am sure that Charizard does better against everyone else. I find that every one of Charizard's bad matchups is also a bad matchup for Bowser, while Charizard does well or OK against some characters Bowser struggles against, such as Greninja and Duck Hunt Duo.
Let me break down why Charizard is generally a superior version of Bowser in 1 VS. 1 by listing the top 5 reasons most important reasons:
1. The outer half of Bowser's Fire Breath does not flinch, unlike Charizard's Flamethrower, where the entire length of the flame flinches. This gives Charizard a superior neutral, zoning and edgeguarding game, which is important in almost every matchup, especially against projectile campers, since Charizard can much more easily approach them by jumping towards them and then utilizing a falling Flamethrower, unlike Bowser. It cannot be stressed enough that this is by far the most important reason why Charizard > Bowser in 1 VS. 1.
2. Against good players who play defensively, Bowser, surprisingly, has trouble landing a KO at a reasonable %, since almost all of his KO moves are very difficult to land with, and often involve punishing someone who approaches Bowser (for example, using Usmash to hit someone trying to fall towards Bowser and attack him from above, or rolling towards him, using Fair or Bair against an approaching opponent, or using Dair or Down B against someone chasing him in the air). However, if the opponent does not approach Bowser, but instead forces him to approach, such as with projectiles, then while he won't have much trouble racking up damage with jumping Fire Breath approaches or grabs, he will have a very hard time actually landing a move that KOs, unless he gets the opponent to very high %s. Charizard, on the other hand, does not suffer from this problem due to his myriad of easy-to-land killing tools, including a kill throw, Flare Blitz, and two amazing smash attacks.
3. Charizard's jab and standing/shieldgrab, two extremely fundamentally useful moves, are both faster and have longer range than Bowser's. Just experience the Charizard VS. Shulk matchup, then experience the Bowser VS. Shulk matchup, and you will see just what a world of a difference the difference in the two characters' jab and grab speed and range makes.
4. Bowser does not have a general roll-punishing tool that is anywhere near as good as Charizard's Dsmash. Bowser's own Dsmash has way too little range to punish rolls anywhere near as effectively, Fsmash requires a hard read (you have to predict the direction of the opponent's movement whereas Charizard's Dsmash can hit anyone close to Charizard in general), and Whirling Fortress does not kill anywhere close to as early as Charizard's Dsmash does.
5. Superior recovery.
Charizard has other advantages over Bowser such as having higher dash speed and a much faster Usmash, though they're not as important as the above five.
In Bowser's defense, by far his greatest advantages over Charizard are his incredibly amazing Whirling Fortress, which may be the single best out-of-shield option in the entire game, as well as his Forward B, which makes shielding a far less safe option for the opponent when Bowser is in the air. Of course, Bowser also has other advantages over Charizard such as not flinching from weaker attacks at low %, being heavier, and doing more damage with each attack in general, but overall, I do not think that any of such advantages, Whirling Fortress, or his command grab make up for all the advantages Iisted for Charizard above, especially the first point about the entire length of Charizard's Flamethrower flinching while the same cannot be said for Bowser's Fire Breath. That in actuality makes an INCREDIBLE difference for the two characters' neutral, zoning and edgeguarding games, and it is a large part of why Bowser struggles with zoning characters so much compared to Charizard. I honestly believe that Bowser could actually be better than Charizard in 1 VS. 1 if only the entire length of his Fire Breath flinched, like it did in Melee and Brawl, as that would VASTLY improve Bowser's neutral game.
With all of that said though, I do think that in 2 VS. 2 with Team Attack on (and it is always on in competitive 2 VS. 2 matches), Bowser is generally better, for four main reasons:
1. Charizard's greatest advantage over Bowser in 1 VS. 1, Flamethrower's superiority over Fire Breath, is nowhere near as important in 2 VS. 2, as Flamethrower and Fire Breath are inherently far, far less useful moves in 2 VS. 2 compared to 1 VS. 1, due to the way Charizard and Bowser both leave their backs wide open to being attacked while using these moves (a consequence that is irrelevant in 1 VS. 1 unless Charizard/Bowser misses with their Neutral B, while this can be disastrous in 2 VS. 2, since Charizard/Bowser can be punished by a running Usmash in the back by one of the opponents while Charizard/Bowser is busy burning that character's teammate).
2. The value of Bowser's greatest advantage over Charizard, his god-like Whirling Fortress especially out of shield, is significantly greater in 2 VS. 2 compared to 1 VS. 1, as an instant "get away from me" option is far more valuable when one has two opponents to deal with. Charizard, in comparison, does not have an equivalent option, as his best out-of-shield option, his shieldgrab, can only target one opponent (and God forbid him accidentally grabbing his teammate), and his Dsmash is far slower than Bowser's Whirling Fortress.
3. The fact that Bowser's KO moves are harder to land with than Charizard's is not as heavy a disadvantage for Bowser in 2 VS. 2 compared to 1 VS. 1, since all attacks are inherently easier to land in a 2 VS. 2, where everyone needs to focus on more characters. And since Bowser's moves generally deal more damage than Charizard's, it is evident that a 2 VS. 2 environment where hits are easier to land rewards him more.
4. One of Charizard's main KO options, Flare Blitz, is far riskier and less rewarding in 2 VS. 2 due to the possibility of accidentally hitting one of TWO characters besides the character one intends to KO with the move (and one of those two other characters is one's own teammate), combined with the fact that Flare Blitz stops as soon as it collides with one character. For example, say Charizard is on the left side of Final Destination, while the other three characters are on the right side - Charizard's teammate, an opponent who has high %, and an opponent who has low %. The character you'd want to hit with Flare Blitz is the opponent who has high %, but it is very difficult to make sure that that is the character you will end up hitting with Flare Blitz, since all three of the other characters can jump, move, and roll around, and God forbid Flare Blitz hits one's own teammate while he or she is at KO %. Also, Charizard dealing 9% to himself in exchange for KO'ing one of two opponents is less worth it than Charizard dealing 9% to himself in exchange for KO'ing one's only opponent. And this is not even mentioning the fact that even if Charizard does successfully land Flare Blitz on the correct target to KO him or her, the KO'd target's teammate can easily punish Charizard when he falls on the ground after the collision.
I am not in any way implying that Charizard and Bowser are not very, very different compared to each other, but as a person who has played both characters extensively, I can confirm that with everything in mind, Charizard generally outclasses Bowser in 1 VS. 1, although I do feel Bowser is better in 2 VS. 2.