You are right that teenager is generally 13-19, though it also sometimes means 10-19 (though the 10-12 range is sometimes called "pre-teen"). Adolescence, however, is a descriptor of a period of life, not a fixed set of numbers.
That means that we can talk about early adolescence (roughly 9ish to 14ish), adolescence (contextual, but usually in the 12-17 range), and late adolescence (maybe 16ish to 22ish). I hesitate to give these numbers at all, because they're very, very rough. I'm just trying to give you a sense of how they're used.
But now let's put that into some more context: first of all, they really are often used synonymously, particularly in informal speech. Folks might well use "teenage years" to refer to years that are not, strictly, in the teens at all. It can refer to puberty in general, or can refer to characteristics that are associated with teenage years (such as rebellious behavior)
And as for formality, you will rarely see teenager written in very formal text, but I think that's more because, when you get into medical journals and such, the strict "teenage" designation is not that useful to speak about. It is not really a phase of life, so there are rarely medical, biological, or psychological characteristics that wouldn't be better identified by "adolescent".