You belong with me.
This refers to someone being in the right place (generally because the speaker and the person being spoken to are in love) when they are with the speaker. This "being together" is of course not necessarily a matter of being physically near to each other, and instead it can refer to 2 people being in a committed (and ideally monogamous) relationship with each other.
You belong to me.
In this sentence, the speaker is expressing "ownership" of the person being spoken to. It's more one-directional, almost ignoring the other person's opinions on the matter.
I guess if we were to read into this a little further, the first sentence implies a sense of "correctness" in the two people being together, as if the circumstances make them suitable for each other, in addition to their mutual attraction. Call it "destiny", if you want to.
The second is definitely more egocentric, stating that the speaker claims (ideally exclusive) ownership of the person being spoken to, and therefore also claiming the right to govern their actions. But this unbalanced power situation also implies that the speaker may also "own" other people (i.e. lovers), and that the person being spoken to is powerless to change that.