Do the prime factors of a number differ depending on which factors you choose first explain

No. If you do prime factorization correctly, there can only be one set of prime factors.

Example:

24→3 and 8→ 2 and 4→ 2 and 2

24→2 and 12→3 and 4→ 2 and 2

The prime factors of 24 are 3, 2, 2, and 2. Hopefully this answered your question.