Should I say:
[•] I'm thinking about someone.
[•] I'am thinking about somebody.
somebody is more "concrete" than someone because it contains the word body and for that reason somebody tends to be used more often in casual conversation, especially if the conversation has a physical component, and less often in contexts that require a degree of formality. someone is neutral and can be used in both casual conversation and more formal contexts.
Somebody threw up in the boys bathroom.
If someone should phone asking for me, tell them I'm in a meeting.
In most contexts, they are interchangeable. The only difference that most native speakers can agree upon is that someone is more formal than somebody (just as anyone is more formal than anybody, and everyone is more formal than everybody).
They are synonymous and interchangeable. If I look them up in Longman Dictionary of Contemporary English (5th version), I get the same definition, however, we know that no two words mean exactly the same (not even couch and sofa).
What's more, if we use Merriam Webster's dictionary for finding out the difference, there might be some.
some person: somebody hoping that someone will suddenly find out - Contemporary Review, then you meet someone ... you like - John Van Druten
one or some person of no certain or known identity: a person indeterminate if you leave the door open somebody will be sure to come in, there should be somebody at the office at this hour
a person of position or importance think oneself a somebody - often used without article, the desire to be somebody is one of the strongest of human motives - American Quarterly
If you consider the abovementioned differences, you might find out which to use.