I can't speak for the AVRs, which I don't use, but I can explain the PICs a bit. There are various families.
The 10F and 12F chips.
These are tiny 6 and 8 pin chips used for the most basic tasks. I did a digital noise gen based on a 12F508. Even with such small chips, the 12F675 has an ADC and some analog inputs, so they're not completely limited to copying basic logic functions in less chips and less space, but that is probably their most useful function.
16F chips
These are basic 8-bit processors. They come in so many varieties to give you some choice over memory size and which peripherals you want/need. Obviously this makes a lot of difference if you're a manufacturer making 1,000,000 units, but isn't so important for us if you're talking about the difference between a chip that costs $1.20 and one that costs $1.27.
The available peripherals include multi-channel ADCs (8 or 10-bit, usually), SPI, I2C, serial UART, and more specific stuff. My personal favourite is the PWM module which can be used as a cheap DAC to produce up to 10-bit output. This will generate audio, but it's not CD quality. It's good for modulation sources - envelopes and LFOs and such like.
Max clock speed is 20MHz.
16F18xx chips
These are a recent improvement on the above 16F chips. They've added some extra instructions which help speed things up, and increased the maximum clock to 32MHz, as well as offering a low voltage version for 3.3V supplies. They're an evolution, but they have a fair bit more power.
18F chips
These are improved 8-bit chips. They include a multiply, which is a notable plus. I've never used them, so I can't comment in detail.
24F/24H/dsPIC
These are 16-bit chips. The 24F are aimed at low power applications, whereas the 24H aim at performance. The dsPICs include some DSP instructions. These are powerful if you can make best use of them, which isn't easy and takes a bit of rearranging. Some of the instructions do five or six things at once though, so it's worth taking the time to learn to use them to their potential, even if it is a bit around-the-houses to get there.
The 30F chips are +5V and tend to be what you need f you want larger DIP format. The 33F chips are 3.3V and use much less power. There are some great 28-pin DIPs available here. My favourite is the 33FJ128GP802, which has a 16-bit stereo DAC on the chip. Ace for all kinds of oscillator projects and audio generation stuff. There are others. For audio *processing* you want to be looking for chips with the codec interface peripheral. This gives you an easy way to get audio ADC/DAC on and off the chip. Seb Francis has some good projects online at
http://burnit.co.uk/sdiy/ demonstrating this.
32-bit chips
Dunno. Haven't played with 'em, so I can't really comment.
For all this, there's probably an AVR roughly-equivalent chip. There'll be some benefits to one and some to other. Ultimately I don't think it matters too much which you use, as long as you're able to make it do what you wanted it for in the first place. I do a lot of synth stuff on PIC and dsPIC, but Olivier Gillet over at MutableInstruments has done many synth things on AVR. Either does the job.
2cents,
Tom