Does armor proficiency from class also give the Ability Score increase from the proficiency feat?

Old school RPer coming back after long gaming hiatus and new to D&D 5e. Liking what I see, and happy to be rid of ThAC0, lol. I’m pretty sure I already know the answer to what I’m about to ask, but figure I’d ask anyways to make sure I didn’t miss something in the new system. Sorry if this is clearer to others or was addressed previously.

As I’m reading it seems feats are a little different now and fewer and further between when leveling, so I’m guessing they’re more valuable than they used to be. Going through the feats list I noticed that a lot of the feats also give an Ability Score increase and I have a question about those increases. (Example: Moderately Armored feat gives proficiency in medium armor and also increases STR or DEX by 1.)

IRC in the old system if you were proficient in an armor type you effectively had the armor feat for it. My question is, does a class in D&D 5e that starts with proficiency in medium armor have the Moderately Armored feat and get the Ability Score increase from above example? Or is the feat totally separate and they can just wear the armor but don’t get the Ability Score increase unless they take the feat?