The OE gas cap has a valve in it, so yes it would matter. Vented went out with the '60s. Here in CA, as part of the SMOG test, the cap is removed and tested to make sure it maintains seal with pressure, but allows air in with vacuum. So yeah, it is a working part that can go bad and need replacing. My local SMOG shop happens to stock replacement gas caps should that be the only thing causing one to fail their test. They're affordable, though I'm not one to recommend throwing parts at a situation without prior diagnoses.
Now you've added some useful information... recent repairs that directly affect the fuel system (and coincidentally are near the fuel smell). Did you use a new o-ring with your work? Can you re-do or double check that stuff? I'll put a link to the fuel chapter of the FSM that you can use to verify you did it all correctly. You might want to pull it out, check your fittings and hoses and such, then put it back, possibly with a new o-ring seal. That would be a starting point before all the other stuff. Though I do recommend a new fuel filter. You're due for one of those.
http://www.nicoclub.com/service-manual?fsm=QX4/2002/FE