I hear both as "I can", but when is it appropriate to use either or?
I am aware the former means "maybe".
The main difference is that могу is a verb, and можно is an adverb. If you confused about the meaning or using these words in practice: могу is more like when you say 'I can' (do something or expressing a state about yourself depending on context, etc), e.g. я могу быть сильным/I can be strong. можно is usually used in the questions or when you are asking for an allowance to do something e.g. можно мне войти?/can I come in? However, in Russian language we also use могу in questions, e.g/. могу я войти?/can I come in? Thus, могу and можно in questions will mean the same
Может быть can be also used as “maybe”
Examples:
Может быть, Вы правы
Maybe you're right.
Может быть дождь до вечера.
It might** rain before evening
Может быть, в этом и есть доля правды
There may be some truth to this
Могу used as Can I/May I:
Могу ли я начать?
Can I begin?
Могу ли я включить свет?
May I switch on the light?
In the Russian verb phrase there are two modals: one of possibility and the other of ability.
So the English equivalent must be:
I may (or might) be able to do...