Practically speaking, few people will notice any difference. However, looking closely at the words, #1 expresses an obligation to pay whereas #2 only expresses an opportunity to pay.

Great question. In the first, "I've got" expresses obligation. You could substitute "I must" for "I've got to." In the second, "I've got" expresses possession. You could substitute "I have." This amount of time is available to me as a window of opportunity in which I can perform the needed action. The preposition "by" has the deadline as its object, and you could substitute "before" for "by." The proposition "until" and its object delineate the available window of opportunity for performing the action, and you couldn't substitute "before."