I have a "first run" RPi and I have found many uses for it as a low power (about 1 watt) headless Linux controller. Running headless you can communicate with it over ssh either with a FTDI UART connection or over the built-in 100mbps wired or USB dongle wireless internet adapters. It is much more capable than any Arduino or PIC for general computation particularly if you need floating or fixed point. The graphics engine can decode Full HD 40Mbps Blu-ray in real time. It is also a fine platform for soft real time use, with analog (A/D) and even PWM. Because it's Linux, you get USB, wireless ethernet and BT, a real file system, robust ethernet networking with SSH/SSL, standard Linux apps, the command line, etc., all without doing any work. It supports SSH/SSL encryption, which Wiznet devices don't do, or at least didn't do when I looked at them a year ago. It even has decent stereo audio which, as LadyAda mentioned on the AdaFruit website, is surprisingly difficult to do on a MCU board at low cost.
For apps that don't require PC-style I/O, the RPi is not a great choice. It's not as good as an Arduino for real time like PWM and such, it has a weird undocumented SOC, and the SD card is dog slow. But for remote sensing / logging / wireless ethernet where you need to process the signal, at $35 it can't be beat. Let's put that into perspective. An Arduino Uno R3 + Ethernet Shield is $75, and an Arduino Uno Ethernet board is $65. I think it would be a challenge to even build an Arduino Uno Ethernet on a homebrew PCB for $35.
I think the closest community supported competitor to the Pi in the market now is the BeagleBone. It has similar memory and performance, but it has a fully documented ARM 7 (TI AM3359 A8 SOC), no "chip on chip" issues, and most every I/O type you could want. It accepts expansion shields ("capes") of which perhaps 10 are already in production and many more on the way. But it is a different class of device, much like the ARM7 boards someone mentioned earlier. At $89 it costs 2.5x as much as RPi, and it uses more than twice as much power.
RPi is not going to replace Arduino or BeagleBone, and it's not going to replace a PC either. But for me at least this is an extraordinarily useful board at a throwaway price. I think it will also meet its original goal of giving kids something they can hack, at a price that won't break Mom and Dad's budget. I agree it is a victim of its own hype, but I think that will soon abate as people figure out what it is actually capable of.