I have both the Wii and 360, and I love 'em both.
The 360 has a much better online experience - most games support XBOX Live and they have loads of downloadable games, demos, shows, music videos, etc. Plus if you have Windows Media Center Edition (or Vista Home Premium) then the 360 is a media center client which lets you see/hear your videos/music/pictures as well as go online to places like VH1 VSPOT.
Both the Wii and XBOX 360 do wireless (but you've gotta buy an adapter for the XBOX360). I ran a cat5 cable to my TV and put a switch in there so both Wii and XBOX are on my wired network - wireless doesn't cut it for serving up HD content from my PC.
The 360 has HD plus a nice cheap HD-DVD drive (which you can now get with 5 free HD-DVDs...I paid $157 for my drive).
The 360 has great lineup of games, many of which don't have Wii counterparts. The Wii has the fun factor all over the XBOX but I think the XBOX has a more intense experience.
As far as the red rings, just keep the XBOX clear of dust and well ventilated and you have nothing to worry about. Its basically a high powered PC without the cooling that a PC with those kind of processors would normally have (to keep the noise down). So some common sense is all it takes.
I got the XBOX first, with games like Tiger Woods golf, Forza Motorsport 2, and Halo 2 (got Halo 3 on preorder). They blow peoples minds when they see them on our 42" Panasonic plasma.
About a month after that we got the Wii. I would say we play about 60% Wii and 40% XBOX. My younger kid loves the Wii, she would play Smooth Moves all day if I let her! My oldest is a Halo freak so he is always on the XBOX360. Me, I split between Tiger Woods on the XBOX and Madden on the Wii.
I love when people ask me "which do I like better" because there really isn't an answer - its whichever I feel like playing at the time. They are different, and each has its own strengths and weaknesses. I am really glad that we got both, they complement each other nicely.