Is there a difference between болен and заболел or they are just synonyms? The context is "I am sick". (Я болен vs. Я заболел)
There's an important difference in meaning. Болен and больной can be used to say "mad", "nuts" while заболел is never used for this purpose.
Ты болен?! — Are you mad?!
Больной придурок! — Sick bastard!
Ты заболел?! — Have you become ill?! (e.g. caught a cold, flu etc., I'm worried about your health)
[Этот] придурок заболел! — This idiot became ill. (e.g. caught a cold, flu etc. and now we have problems because of his absence)
Grammatically, болен is a noun on a bordeline between a Participle and an Adjective (from больной / болеть and in this sense it is close to verbal nouns in English) and заболел is a perfective verb (заболеть).
The difference is personal attitude / modality, as Alissa has already pointed out.
The other difference is that болен (as most of other shortened adjectives) are used for closing a topic, whereas заболел suggests further communication.