Does food cooked in firewood taste different than that which is cooked in gas?
"Taste better" is entirely opinion based and unanswerable with anything but opinion.

As to does it taste different, absolute.  How different depends on application and technique.  Different woods taste and cook differently.  Some have high resin, some a harsh smoke, some a more gentle smoke.  They burn at different temperatures and different moisture.  Some may have toxins in there smoke.  One would not use a green oily wood and expect something palatable.  Smoking food is its own artform and people spend a lot of time trying  to match correct woods to specific applications and even that is largely opinion.

Left out non-proteins:  Certainly the same rules apply and it remains a matter of taste.  Some foods grab more of the potential flavors, some less.    One is not even limited to solids as soups could be done too.  As bob1 points out, there are ways to reduce the differences in taste, like making sure the wood is burned down to coals with no visible smoke.  I would also say you would want heat to be indirect and you may still have issues with differences in humidity, but you can certainly get at least close to oven flavors or the flavor of a gas grill if that is your goal.