I agree mostly with they others that have posted. You really need to try both packages out for yourself and see what fits best for YOU.
We actually use both. If character animation is what you are after, then I think Maya is the way to go. Max's bones wig out sometimes. As far as the IK goes, I actually like Max's HIIK solver better. Both have the crappy ON/OFF switch for ik that is useless.
For dynamics and particles, Maya wins hands down. Reactor is pretty nice and really fast, but the integration just isn't there yet.
Modeling. This is a tough one. On one hand Maya has the best NURBS out there, but I mostly do chaaracter work, so I try and stay away from them. I like Max's new poly system with the Meshtools better than Maya tools. There are some scripts out there that replicate what Meshtools do, but I find the workflow of Max's to be more like Miria.
Materials/shaders. I think I like the hypergraph better than Max's material editor. The reason is that it is easier to see whats going on when you start to have a complex shader. I must say that I think BOTH could usse some work here.
Rendering. Hands down, Max. It has it's problems, but Maya's shadow maps are nasty, and the renderer is way slow. speed is a big deal when you don't have 100 cpu's behind you.
Also, a couple of others were discussing PRMan for Maya. Max also has hooks into it through MaxMan. You also have Exluna, Final Render, Vray, Brazil, Arnold?, Mental Ray, and a few more i'm sure.
Well, I tried to be as fair as I could about what I have found to be in both packages. Whatever you are wanting to do, I'm sure you will be able to pick the best package for you. Both are great, and both make you angry every day.
Hope this helped.