Quick answer
мочь = able in this situation. уме́ть = have learned how. мо́жно = is it allowed?
Я уме́ю пла́вать, но сего́дня не могу́.
I can swim, but today I can’t.
мочь
able right now
- This situation, this moment
- Circumstances allow it
- Often in the negative
- + infinitive
Я не могу́ прийти́.I can’t come.
уме́ть
have learned how
- A skill you acquired
- Independent of today
- Swimming, cooking, driving
- + infinitive
Она́ уме́ет рисова́ть.She can draw.
мо́жно — the third one
Мо́жно asks about permission and needs no subject at all. It is the politest way to ask for anything.
Мо́жно счёт, пожа́луйста?May I have the bill, please?
Мо́жно войти́?May I come in?
Здесь нельзя́ кури́ть.Smoking is not allowed here. — нельзя́ is the opposite
All three in one sentence
Я уме́ю води́ть, но сейча́с не могу́ — я вы́пил.I can drive, but not right now — I have had a drink.
Мо́жно здесь припаркова́ться?Is parking allowed here?
Typical mistakes
Я могу́ пла́вать. — meaning “I know how”
Я уме́ю пла́вать.
мочь would say the conditions allow you to swim today.
Могу́ я счёт?
Мо́жно счёт?
Asking permission in a café is мо́жно — shorter and far more natural.
Check yourself
Which sentence means “May I come in?”
Permission is мо́жно. The third option is understandable but sounds like you are asking about your physical ability.