I now own both of these players, and for anyone interested here is a head-to-head opinion. Both of these I used connected to the same plasma (a Panasonic TH-50PZ81B and running through a Sony STR-DA2400ES amp.).
First of all the interfaces. The Sony on-screen GUI is more user friendly than the Panny and easier to navigate around. The Panny interface is fairly basic in comparison.
Start-up times for the loading of a Blu-Ray disc seem marginally quicker on the Sony, although there is not much in it.
Out of the box, the Sony required an update to make it profile 2.0 compliant whereas the Panny worked straight away. After the update, the Sony refused to operate BD-Live on my Casino Royale disc. The message I got was 'Not enough space, try deleting some files'. I did not have that problem on the Panny.
On standard definition DVDs I couldn't see any difference. Both of them made an excellent job of playing them and I really wouldn't want to say one was better than the other. On one particular film though (Phantom of the Opera) I did notice that in a couple of scenes there was a bit of flicker on the background scenes on the Sony that did not appear to be there on the Panny.
When playing HD both players were fantastic and I couldn't really rate one better than the other, although the sound on the Panny seemed to have a bit more bass to it (but that could just be me). I did set the sound output to direct so that my amp. did the processing (The Sony does not support DTS-HT Master Audio on-board whereas the Panny does).
The biggest difference I found betweent he 2 was the type of discs they would accept. All the standard purchased discs (both standard DVD and Blu-Ray played without problem), but when I tried some personally recorded DVDs (family home videos put on DVD), the Sony refused to play them and all I got was a message saying 'This type of disc cannot be played'. The Panny, however, accepted them without problem. I then tried a copied DVD (Legal I hasten to add) and that again would not play on the Sony but did in the Panny so it seems the Sony is quite fussy over what type of discs it would play.
Styling wise I did prefer the look of the Panny, it just looked a lot more expensive than the Sony.
I got both my players as part of a package (I bought a Sony LCD for upstairs which came with an S350 and a Panasonic Plasma for downstairs which came with a BD35), but if I was buying one for myself on it's own, then I would probably go with the Panny just for the reason that it plays a wider range of discs and the looks of it. Plus, in my opinion, the sound was slightly better. I must admit that I prefered the remote control on the Sony, it just felt more comfortable to use.
Incidently, both players worked over the Viera Link and Sony Bravia link so I could operate both from the TV remote control no matter which TV they were connected to.
Hope this helps.