Tolerance, Acceptance, and Approval
The distinction is often blurred by people when they're trying to slander a group1 or hide thier real feelings.
It's not just a semantic difference; this distinction is having a very particular and specific effect on my life, right now. (I'll tell the story later, maybe.)
So, here's the definitions.
Tolerance: to allow the existence, occurrence, or practice of (something that one does not necessarily like or agree with) without interference.
Accept: to believe or come to recognize (an opinion, explanation, etc.) as valid or correct.
Approve: to officially agree to or accept as satisfactory.
Accept and approve can be a little blurry between the two sometimes (and there might be regional connotations), but if you're expecting to treat them as different, you can easily find out which is being used in which way.
These sorts of things are easier in practice, so let me give you a musical example...from my workplace.
I tolerate listening to Justin Beiber at work. I don't like it. I don't think he creates particularly good music, and in general don't like anything I've heard of him as a person. But I don't run screaming from the room... though I do question the musical taste of someone who does like him.
I accept listening to Rhianna at work. Some of her songs I like, some I really like, and some are just okay, but I can definitely see the appeal of her work, even for the songs I don't particularly care for.
I approve of Bruce Springsteen at work, particularly from the Born In The USA era. This album is totally acceptable and completely satisfactory to me.
Of course, your opinions of what music and albums are intolerable, tolerable, acceptable, and approvable will vary, but that's not the point. When people say that they want others to tolerate their lifestyle, life choices, partner(s), and so on, they're not asking for you to agree with them.
They're asking you to allow the existence, occurrence, or practice of (something that one does not necessarily like or agree with) without interference.
They're asking you to be civilly polite.
If that's too much for you to deal with, then that's your problem, not theirs.
1In my experience this has always been the political right slandering the political left - when searching "intolerant liberals" and "intolerant conservatives" on Google images I was presented with nearly identical results which slammed liberals. For a thoughtful discussion of whether or not liberals are "tolerant", I recommend "Intolerant Liberals" by Tucker FitzGerald. The most I see the political left doing is saying "tolerance" when they mean tolerance and "acceptance" and "approval".