No, it doesn’t necessarily have to be the difference in subjects (i.e. You or I).

잘 쓸거예요 will directly mean, using “I” as the subject, “I will spend (it) well.” Where as 잘 쓸게요 will also refer to the future, but more in the way of “I SHALL spend it well,” and this form is very commonly used after receiving something (that you can use or spend) as a gift and basically telling the person “(Thank you) I’ll use/spend it well.”

It’s more in the idea of learning new information and then making a definitive decision about the future, saying something like, “Okay, if that’s the case, then I shall do this.” Does that make sense?

I know this concept is a little bit difficult, especially since the “shall” (in N.American English anyways) is almost completely lost, but yeah, for this particular example, I’d focus on the fact that it’s kind of a formal response after receiving a gift.

Yeah, I may have to make a video for this one, bit of a difficult concept.