Thanks for your help¡ =D It was so helpful

I'd say the simplest way to describe the difference is that "to be able to" gives no implication of difficulty in the task, wheras "manage to" implies doubt of success. Some examples: I was able to get a ticket to the concert. (This means I called the ticket office or checked online, they had tickets, and I bought one.) I managed to get a ticket to the concert. (This means I called expecting that there might not be any tickets left or that they would be too expensive, but there were tickets and I could afford them, so I bought one.) I will be able to catch the next train to Madrid. (This means I arrived on time, bought a ticket and when the train arrives, I will be on it.) I will manage to catch the next train to Madrid. (This means that even though my taxi may have dropped me off late or even though I may have lost my ticket and had to buy another one, I am still going to be on the next train to Madrid.)

If you are able to do something, you have the ability to do it if you want to. You might be able to do something, but you don't have the time, or just don't feel like doing it. If you managed to do something, you got it done, even though you did not have much time to do it, or it was a hard thing to do. However, some how, you managed to get it done.

'To be able to' is something, you can do, now and in the future....'I am able to speak English,' I can do it. 'Managed to,' is in the past tense. Something which you did that was difficult, perhaps you thought you could not do: 'I managed to pass my exam...' 'The mountain was high, but I managed to climb to the top.'