It depends on many circumstances. I main Falco, but I often switch between the two.
The different circumstances for me are:
Who am I playing?
Which character do they use?
What stage are we playing on?
How warmed up am i?
How comfortable do I feel with my characters at that point?
I main Falco, and I most often start the match, regardless of who I play against as Falco. However depending on many things, such as who I'm playing against, and which character they use, I may switch between Fox and Falco. Stage choice makes a huge difference, too.
I'm comfortable on all neutral stages with both Fox, and Falco, however depending on their character, my preferences changes. For example:
Against Marth, as Falco, I'm the most comfortable with Dream Land, and Pokemon Stadium, with Yoshi's Story and Fountain of Dreams up next. I have the most problems on Battlefield and Final Destination, with Final Destination being my absolute most hated stage. This rule applies with any character that can chain grab Falco.
against Marth as Fox, I'm the most comfortable with Dream Land, and Pokemon Stadium, and Battlefield, and Fountian of Dreams close by. I have the most problems with Yoshi's Story and Final Destination.
Now for counter-pick levels.
On Mute City, and Kongo Jungle 64, I prefer Falco. With Fox, Green Greens is without a doubt my favorite counterpick level, though in reality I don't mind any of them. I especially enjoy Jungle Japes, and Pokefloats. I have a really gay play style. My play style is focused around playing as efficient as possible, and as gay as possible, focusing on gimps, shine spikes, and edge guarding. Green Green, Jungle Japes, and Pokefloats are the three counter pick stages that Fox excels at playing as gay as possible, and eliminate many of the problems I have versus characters such as Marth.