actually it's simpler than that, in general cases we use ""could"" to describe abilities that we had in past tenses or to imagine hypothetical abilities in conditional cases, and it's so usable in spoken language, in the other hand ""able to"" is so similar but more flexible because we can use it in future tenses for example, which we cannot use either "can" or "could", so we are forced to use "able to". and that's the only observable difference

Between "could" and "was able to" in the past, we normally use "could" in a general sense, and "was able to" for a specific point in time. If you manage to do something, that means you finally succeed, but with some effort. To succeed in doing something (note the -ing form) pretty much has the same meaning, but it doesn't imply the effort that "manage" does.

Could is used to indicate the possibility of something, for example: You could fall if you run down a steep hill. It is also the past tense of “Can”. Could is also used to request something in the present tense, usually in question form. For example: Could the girls come with me to the store? Could you pass me the gravy? or Could you please go walk the dogs? Hope this helps.