The first one means more of "let me know if you pass/do well on the driving test". The second one is more like saying "let me know if you finish the driving test."

I concur with all of the above, with one clarification: "get through" to my ears denotes a slightly weaker level of completion than "succeed" or "successfully pass" -- it can often/sometimes carry a connotation of "survive" --- ie, someone might say something like "That exam was really hard; I got through it, but I doubt I did well."

No...the meanings are very close but not exactly the same. "Let me know if you get through the driving test." suggests that the driving test is an ordeal (difficult experience) to survive or "get through". The implication is "Let me know if you pass the driving test." "Let me know if you get through with the driving test." doesn't have quite as much of a "difficult experience" implication. I sense that it's more of an "annoying experience" that has to be "gotten through with". However, in the case of a driving test, the only way not to "get through with" it would be to give up before finishing. Since the listener would fail the driving test if he gave up before finishing, the end result is the same as in the first sentence. After writing the above, I realized that the difference between the meanings of the two sentences is subtle and, in this particular case, there is not that much difference. However, "get through x" and "get through with x" do have different meanings so it's worth paying attention to the difference in other situations. So.. an example of "to get through with" is that the driving examiner might start the day and say, "Oh my, today, I have to get through with 10 driving tests before I can go home. I hope the applicants are not too incompetent." (NOTE: This example sentence is a bit forced but I am trying to use the phrase "get through with".) Another example, a job applicant might hope to get through an interview (successfully). The hiring manager might feel he has to "get through with" six interviews in order to pick the right job candidate. (NOTE: the hiring manager might also say that he "has to get through" six interviews.) English is a strange and annoying language with subtle differences like this. Hang in there. Good luck.

Frank, Although very similar, the two sentences do have different meanings. "To get through a test" implies that someone has successfully passed an exam. "To get through with a test" implies that someone merely completes a test but the results are unknown. Additionally, you are correct that if the above sentences are changed from "get" to "got" as the tense change indicates the test was already taken in the past versus the future.