Axler's book is much more of a pure math book, in the spirit of an abstract algebra or analysis book. Strang's book is more applied, applications oriented, more mechanical in nature. Both approaches have their place, Strang is more suitable for engineers. I find Axler far more elegant, and his arguments generalize much more readily to infinite dimensions. I would recommend Axler for budding mathematicians and people wanting to study functional analysis and operator theory later on.