嬉しい (うれしい) means that something you hoped for happened, which is why you're happy/satisfied.
幸せ (しあわせ) on the other hand is more general like the English word happy. It means having good "fate" or being in a good state.
So it really depends on why you want to say you're happy. On the other hand, the words are often interchangeable since if something you hoped for happened, that could mean you have "good fate" or whatever.
To clarify, しあわせ is much more serious and long-lasting than うれしい. For example, if get a couple hundred dollars back on your tax rebate, you may be うれしい, but if you find your "soulmate," or move to Japan and decide you love it and want to stay there forever (or at least long term), you will be しあわせ.
better off using うれしい as saying ''I am 幸せ'' is like saying ''I am good fortune.'' Not sure if you would say use もらう for the latter, as in ''I received good fortune'' but I think しあわせ is more of statement made about an inanimate object--not necessarily an abstract mood. depending on the context, too, and also how happy you are, I'd throw a チョウ in front of that too. CHOU URESII as in ''I'm stoked!'' (but again, depending on context: a slang situation)
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