Yes, gameplay differs by character! And in some cases, dramatically. There are plenty of powers associated with cards, but some things are unique to specific characters. Karlee, for instance, lights up all hazards and special enemies. No one else can do that, which makes her uniquely suited for scouting or hunting special enemies.
Some characters help specific builds. Holly’s the obvious choice for a melee build, but then again Evangelo gets free breakouts when he’s grabbed by enemies, which is a constant peril for melee builds (everyone else has to use an inventory slot to tote around stunguns, but Evangelo can’t carry something else instead). If you want to be a dedicated healer, which is probably necessary on harder difficulty levels, you want Doc for her free healing and inherent healing bonus (other characters need cards to heal efficiently). To flex consumable weapons like grenades and pipebombs, you want Hoffman. Mom’s free revives are probably a godsend on harder difficulty levels. And each character also gives the team a different bonus.
But, yes, the characters definitely play differently.