Like basically everybody else I teach out of this book, and I'm happy to see a new edition. I'm curious what's changed/added -- I already am unable to get through the whole thing in a semester.
At our school students take a computational linear algebra course first (with a lot of row reduction). So I am slowed down a bit by constantly trying to help the students see that the material is really the same thing both times through. I do wish there were a little more of that in Axler.
Sure, I am very familiar with them -- I actually TAed 18.06 for Strang once upon a time. They're great books too. Which is better is mostly a question of what point of view you're after -- if you want to actually calculate anything, Axler's book is not going to help you, but if you want a more conceptual view of the subject it's best place. If you're really serious about learning linear algebra, you probably want to read both, first Strang, then Axler.
At our school students take a computational linear algebra course first (with a lot of row reduction). So I am slowed down a bit by constantly trying to help the students see that the material is really the same thing both times through. I do wish there were a little more of that in Axler.