Carnitine is an amino acid which as a process the body itself does, interconverts between the acetyl ester form and the free amino form, carnitine itself.
Carnitine itself is the active species at least so far as mitochondrial energy production goes. (I have a pretty strong expectation that acetyl-L-carnitine itself has CNS stimulatory properties at higher doses, as this occurs quite rapidly after taking it without time having been available for de-acetylation; and it could be that there are other activities of the acetyl form.)
Acetyl-L-carnitine is better absorbed if I recall correctly (not 100% positive on that, actually the bioavailability is low for any form) and crosses the blood-brain barrier better.
If taking only one form, acetyl-L-carnitine is the better form to take.
Carnitine tartrate is not an ester, but a salt. What this means is that there is no covalent chemical bond between the carnitine and tartrate parts, but only a chemical charge tending to hold them together.
Whether the carnitine and tartrate stay together through the absorption process or its just equivalent to taking carnitine and tartaric acid (which would be a pointless addition) I don’t know.
I would prefer acetyl-L-carnitine over carnitine tartrate, but it’s possible there might be an advantage of carnitine tartrate over carnitine itself. I haven’t looked into it.