Yes, both can be used as the answer to thank you. In this context, they are totally the same. However, the phrase does subtly differ from each other. 别客气‘s recipient can’t be the speaker itself, while 不客气 could. Thus, 我不客气了 make sense while 我别客气了 sounds weird. Also 不客气 sometimes carries a pejorative meaning. As for example in "你还真不客气啊“ (You really are helping yourself!), here the speaker is being sacarstic as somebody has acrossed their personal boundaries.

I just wanted to add that you may also say 没事儿，不[用]谢，甭客气 and 不会. The last one seems to be very common in Taiwan.

They have the same meaning.You can use either one.