CAN
She can speak Spanish, but she can’t speak Italian.
He can analyze people’s handwriting.
We can’t afford to pay the bill.
Helen can read well, even though she’s only three.

СOULD
1. когда что-то мог:

Their son could swim before he could walk.
We could hear a cat, but we couldn’t see it.
Mozart could play the piano at the age of five.

Обратите внимание, что именно ‘мог’, а не ‘смог’.

Для ‘смог‘ используется was able to:
Mike’s car broke down, but fortunately he was able to repair it (смог отремонтировать…; NOT… but fortunately he could repair it)
The plane was able to take off at eleven o’clock, after the fog had lifted (смог взлететь…)
Luckily, Mark was able to get the work done in time (смог выполнить…)
The drivers were able to stop before they crashed into each other (смогли остановиться…)

2. ИЛИ не мог:

Mozart couldn’t speak French.
Peggy couldn’t swim when she was ten.
Tom didn’t get the job as he couldn’t drive a truck.

3. could, обычно, предпочтительнее, чем be able to:

– в отрицательных предложениях:
I tried to get up, but I couldn’t move.
It was too far from the station, and he couldn’t go there by foot every day.

– с глаголами, передающими ощущения: feel, hear, see, smell, taste
She could feel the spray on her face as the boat raced through the water.
He could see the sun appearing from behind the horizon.
I could hear a strange noise coming through the open window.

– ментальными глаголами: believe, decide, remember, understand:
I could remember the crash, but nothing after that.
When I saw them together, I could hardly believe my eyes.
At first, I couldn’t understand what was going on.

– после the only thing / place / time и после всего, означающего ‘the only thing’:
All we could see were his feet.
The only thing I could do was sit and wait.
It was the only place where I could feel at home.

– c almost, hardly, just, nearly:
I could nearly touch the ceiling.
She was so exhausted she could hardly speak.

BE ABLE TO (быть способным)
Используется вместо can / could, если:

1. способность вызывает удивление ИЛИ подразумевает преодоление какого-либо затруднения:

Despite his handicap, he is able to / can drive a car.
Helen is able to / can read well, even though she’s only three.
Is the child able to / can the child tie his shoelaces without help?

2. Если can / could грамматически невозможны:

Will I be able to speak fluently  by the end of the course? (with the future aspect)
If the snow carries on like this, very few people will be able to get to the concert.
When the new road is built, I’ll be able to drive to work in under half an hour.
I haven’t been able to drive since I dislocated my wrist (with the perfect aspect)
We love being able to talk the local language (with -ing forms and infinitives)
We were very lucky to be able to live in the country during our childhood.

3. Если упоминается единичное действие в прошлом и это действие:

– удалось – was able to
We had a flat tire, but we were able to fix it and go on (NOT … but we could fix it …)
Возможные варианты:
… but we managed to fix it…
… but we succeeded in fixing it…

– не удалось – wasn’t able to / was unable to ИЛИ couldn’t:
The lawyer couldn’t (wasn’t able to / was unable to) persuade the jury of her client’s innocence. 
We had a flat tire, and we couldn’t (weren’t able to / were unable to) fix it.
Возможные варианты:
… failed / didn’t manage to persuade… (=didn’t succeed in persuading)
… failed / didn’t manage to fix…(=didn’t succeed in fixing)

Be able to не употребляется:

1. если что-то происходит в момент говорения:
Watch me, Mum, I can stand on one leg (лучше, чем …  I’m able to stand on one leg)

2. перед пассивом:
This book can be used by complete beginners (NOT This book is able to be used by,,,)
Films can now easily be streamed online (NOT Films are now easily able to be streamed…)

3. если имеется в виду «знать как», «уметь»:
Can you cook? (пучше, чем: Are you able to cook?)