Both of them have a meaning of "can / be able to", but "られる" has also some different meanings. Therefore, whenever you see "られる", you have to think what "られる" means in the context. Here are some examples: 彼が言ったことは信じることができる。　I can believe what he said. 彼が言ったことは信じられる。　I can believe what he said.（The same meaning as the above.） 彼が言ったことは信じられている。　What he said is believed.（"られる" is used to make this sentence passive.） 先生は彼が言ったことを信じられている。　The teacher believes what he said.（"られる" is used to sound polite. It is a kind of honorific expression.）

In many cases, the usage of られる/れる/る (when the original form of the verb ends with る, mainly られる is used.) is the same as ことができる. The following are examples: 私はトマトを食べることができる。 私はトマトを食べられる。 Both mean “ I can eat tomatos”. このトマトは食べることができる。 このトマトは食べられる。 Both mean “This tomato is edible”. But られる/れる can be used to express your respect for the subject person: 田中先生はお酒を飲まれる。 Prof. Tanaka drinks alcohol. In English it’s hard to express the respect shown in Japanese verb’s conjugation. The following are also expressions related to capability. 田中先生はお酒が飲める。 田中先生はお酒を飲むことができる。 Prof. Tanaka can drink alcohol. ここではお酒が飲める。 ここではお酒を飲むことができる。 Drinking alcohol is permitted here.

HI Vivian, a hard question.. I would say.. yes, there is a difference. the regular form (..eru) is called the potential form and is used for simple possibilities, like 'I can play the piano', 'I can go tomorrow' or 'Can I pay by credit card?' If you use the other thing (koto ga), it sounds too formal or more serious, like the possibility of doing something doesn't exist. It's not usually used for those simple things in my experience, except if you want to say that something is really possible or not (because just saying 'I can/can't' can mean that you don't have the chance to do it)