I have both a Solo and 350 on my network right now. I started off with a Solo and got my 350 last year after they came out.
The biggest advantage as Kamakzie said, is the 350 is capable of streaming in HD. It's SD streaming is also much better compared to the Solo; the Solo's video tends to look very "rough", as if it's not getting any video processing. I used to attribute that to the SlingPlayer PC app not using my video hardware well; but I stopped using it as the 350 will only stream to a browser, the solo still looks rough in the Chrome player.
I also believe the 350 has built in IR emitters on the sides; allowing for the same pheonomon as when your regular remote control bounces off a wall enough to trigger commands. I still have the IR emitter I got with my Solo stuck to the box, so I haven't had to test the built-in IR. It also comes with a pyramid shaped wired IR emitter, rather than the double "hang-over" type that came with the Solo.