Why would this be specific to California? This almost looks like the new setup with the NREMT except I thought they retitled First Responder to Emergency Medical Responder (EMR). A more specific Californian question would be what's the difference between an EMT-I and EMT-II, which EMT-I is an EMT, and EMT-II is an Advance EMT (AEMT). EMT used to be EMT-Basic, AEMT used to be EMT-Intermediate. I am guessing this changed to the setup you said because we finally switched to a state license instead of a county license (or cert, or whatever anybody wants to call it *shrugs*).
I like to think of it like this, an EMR knows how to execute most of the skills an EMT does, and even though an EMT doesn't know much about medicine (sorry guys, don't wanna hate), the EMR knows less about medicine compared to the EMT.
As EMR is to EMT, AEMT is to Paramedic. An AEMT knows how to execute some of the skills the paramedic can do, however, they know less about medicine than the paramedic does.
Even though both the EMR and AEMT can do some of the same skills as the level above, since they know less, they can only half-ish (or less) execute it like an AEMT won't be able to administer all of the same drugs as a Paramedic.