I think the best way to describe alpine versus millenium silver is that alpine is more of an icy silver with a hint of blue and millenium silver is more subdued with a hint of smoke or beige. Here's a picture I took a while back at a Lexus dealership. The difference is very clear in this picture:

 

I went from a '98 to a '01 both silver and love both colors. Its neat seeing the difference w/ them side by side like that.

 

I always thought the millenium silver had a tiny bit of green in it.

 

I would say the following:
Alpine silver = saturated with silver
Milennium silver = silver with copper tone mixed. It's a unique color.

 

I would agree. People have different taste. While some prefer the alpine, I actually went out of my way to avoid it, and get the millennium. Blends well with the new green tinted headlight housings.

 

gotta love alpine silver