I don't think I would say "completely change" the sound quality. In my experience with the two external headphone options I've tried, they have sounded considerably better, especially in respect to the low end response. I wouldn't break the bank on this, but if you have the funds available go for it, and do some A/B tests.
I thought the Axe 2 already had top-shelf DACs for both analog outputs?...
It does. It uses better converters than a Metric Halo. That said, many converters color the sound and people interpret that as "better". The Axe-Fx DACs are designed to be completely neutral.