Is it possible to mapping reduce either of these languages to the other?
I'll assume that by $L(M)$ you mean the set of strings on which $M$ halts.

Hints:


INFINITE to AALL: Given a Turing machine $M$, construct a Turing machine $M'$ that on input $x$ halts if and only if $M$ halts on some $y \geq x$. (Arrange the inputs in some arbitrary way of your choice.)
AALL to INFINITE: Given a Turing machine $M$, construct a Turing machine $M'$ that on input $x$ halts if and only if $M$ halts on all $y \leq x$.


Both directions require use of dovetailing.