Объясните пожал-та разницу между CAN и BE ABLE пожалуйста. С примерами если можно.

Can – это модальный глагол, переводится «мочь». Основное значение: умственная или физическая способность, возможность (есть и др. значения). Имеет форму прошедшего времени could.

John can speak English very well. – Джон может говорить по-английски очень хорошо
My sister could read when she was 3. – Моя сестра умела/ могла читать, когда ей было 3.
Be able to – это эквивалент модального глагола can, восполняет его недостающие формы, может употребляться во всех временах: Present, Past, Future, переводитья «быть способным»

He is able to do that. – Он способен это сделать. (Present)
He was able to explain it. – Он смог объяснить это. (Past)
They will be able to come. – Они смогут приехать. (Future)

Примечание: в Past употребляется was/ were able to – если речь идёт о единичном действии (a particular ability on one occasion in the past)

Could употребляется для обозначения общей способности (general ability) в прошлом.

Can is used in the present tense. It is used to talk about our ability to do things.

Examples

I can swim.
She can speak English well.
I can swim across that stream.

Be able to is also possible in these cases; however, it sounds a bit more formal here.

I am able to knit. (Less natural than ‘I can knit.’)
She is able to speak English well.

To talk about our past ability, we use could. Was/were able to is also possible.

Study the examples given below.

She could read when she was three. OR She was able to read when she was three.

Again, in spoken English, we are less likely to use was able to.

As you can see in all of these sentences, we were talking about general ability. Things that we can or we could do at any time in the present or in the past.

To talk about things that we managed to do on specific occasions in the past, we cannot use could. Instead, we use was/were able to. The verb managed, succeeded (in…ing) are also possible in this case.

I was able to get some really good bargains in the sale. (NOT I could get some really good bargains in the sale.)

After climbing for several hours, we managed to get to the top of the mountain. OR After climbing for several hours, we were able to get to the top of the mountain. (NOT … we could get to the top of the mountain.)

Both could and be able to can be used to say that we were not capable of doing something on a specific occasion.

In spite of climbing for hours, we couldn’t get to the top of the mountain. OR In spite of climbing for hours, we weren’t able to get to the top of the mountain.

Полностью согласна с Ириной Шиляевой, хочу лишь немного добавить. 1.Could - общая способность, умение выполнить действие: After a few month on the training course,I could speak English quite well. - После нескольких месяцев учебных курсов я МОГ говорить по английски довольно хорошо; was/were able to - УДАЛОСЬ (managed to) выполнить действие в определенной ситуации: He was able to(managed to) swim in spite of the broken arm. - Он СМОГ (ему удалось) плыть несмотря на сломанную руку. Couldn't можно употреблять в любом случае: Fred played very well but he couldn't beat Jack. - Фред играл хорошо, но НЕ СМОГ победить Джека. 2.Can/could,а не able to используется с глаголами чувственного восприятия и умственной деятельности (see,hear,smell,taste,feel,etc):I could understand everything she said. - Я смог понять все, что она сказала.

просто "can" - более распостраненный вариант
можно говорить и так и так, не ошибешься