Попасться может рыба на крючок, в сеть.
Попааался! - can be said by somebody who catches you, e.g. police.
But, which is curious, it is a verb in the active voice. It implies some kind of activity of the subject one comes across or catches, though it may be not deliberate.
So, another person is catching, but it's the victim's responsibility.
Наткнуться можно на что-то самому.
The first meaning is наткнуться на что-то острое.
Наткнуться на нож, наткнуться на ветку дерева в лесу.
Я листала журнал и наткнулась на статью как раз на эту тему.
I came across the article suddenly, but it's me who did it.
So, yes, you are right, but in the meaning of coming across some information,finding something or somebody occasionally "попасться" is more neutral, and наткнуться is more colloquial. It is more polite to say in a formal conversation "если вам попадётся", than "если вы наткнётесь на".
I can guess, that it is so because of the embarassing and awkward situation in the first meaning of the second verb.