There is one case which is simple : subject + pronoun + rappeller
If subject and pronoun are not the same person then it is equivalent to remind.
- je te rappelle que tu as des devoirs à faire (I remind you, that you have schoolwork to do)
- nous vous rappelons que la fête aura lieu samedi (we remind you that the party will be on saturday)
When it is the same person (reflexive form), se souvenir and se rappeler are mostly the same and equivalent to remember. In french, there is no distinction like there is in english with recall (to remember + to talk about it). As Julien said, the major difference is that se rappeler is mostly transitive (*) while se souvenir is intransitive.
(*) As always there are exceptions : je me rappelle de toi, de lui, de vous, etc.
But you should be aware that almost everyone says je m'en rappelle instead of je me le rappelle, and other instransitive constructions with se rappeller are also very common. Purists will castigate the infidels, but what makes a living language alive is that it evolves, so don't be too afraid of making the mistake too.
Regarding the intentional vs spontaneous discussion, I'm not really convinced. To me they really are synonyms.
I would only consider a real difference when it comes to imperative, where in general they are not intertchangeable because the meaning is slightly different.
- Souviens-toi de ce jour ! = you'd better never forget this day
- Rappelle-toi cette leçon ! = you'd better learn this lesson