1) Lax leaves significant results to the problems, and the result is that there is little explanation of their importance.
2) Lax prefers short, mathematically impressive proofs which leverage the mathematics he has developed in the book to that point. But the proofs do not help in understanding.
3) Lax relies heavily on determinants and characteristic polynomials to prove major results. Axler deliberately avoids them, and he says why in the introduction too!
Axler and Lax have two diametrically opposed approaches to linear algebra pedagogy. Neither is "wrong" but in my experience Axler's is much better for learning and Lax's is better for a deeper understanding (and for a different approach).
Not really on topic, but I would like to use this opportunity to express gratitude to the HN community - although this forum is mostly tech-centric, it is wonderful to see things like this post get interest and discussion.