How would you distinguish "having ability or permission" from "having competence for".
Having ability and having competence both mean the same thing. Having permission is unrelated because it means being allowed to do something regardless of whether or not you have the ability/competence. So, whilst you may have permission to play a game, you might not know how to play it - ie you lack the ability/competence. Although, you may only have permission to do something because you have the ability ie I have permission to go swimming because I can swim ie you have the ability to swim.
If I say, for example: "I wasn't able to arrive on time last night" e "I couldn't arrive on time last night", is there a difference? Would you say one conveys the meaning of "impossibility" while the other means "I wasn't competent enough to get there" - maybe the road wasn't good, I had no map or I got lost...? Help me out!!!
Both mean the same in this context of timing. There will be a reason that you didn't get there on time last night, so whether you explain that by saying "wasn't able to", or "couldn't" doesn't really matter.
Or "I can't play tennis" and "I'm not able to play tennis". (as if in the 1st case the person does not have talent for it, while in the second he isn't able cause he's never tried???) Geez, I'm confused!
In this context, it depends on what follows. The statement "I can't play tennis" on its own, means that you do not have the ability to play tennis. However, if you said "I can't play tennis because I need to go to work", means that you do have the ability to play tennis, but you need to go to work instead.
Does this help?