An infected PC *might* stay infected if the botnet owner just has the PC hit this or that server once in a while.
But running the CPU and/or GPU full bore? Guess what -- that's going to lead the the AOLer saying, "Hey, my PC is really slow!" and they'll have their grandson wipe the machine with a fresh copy of Windows XP. When a machine is formatted/wiped/re-installed, the botnet loses the machine.
Then the botnet owner has to get Mr. Novice PC User to click on a "Click here to restore access to your Paypal account." e-mail again. If he doesn't fall for it a 2nd time, that PC is off the botnet for good.
Matthew
You misunderstand the concept of a botnet and viruses/malware.
Good botnet controllers and malware writers do their best to make it so that their meddling isn't detected by the average user. While we make check our router logs and notice something is up, or watch CPU usage spike randomly and realize something is wrong, the average user won't. In terms of bitcoin, all the botnet controller has to do is set the miner to a low aggression so that your average user will not detect it.
And about the paypal scams and such, those are usually just emails sent out by the botnet. Your computer doesn't have to be on the botnet to be subject to such spam attacks.
I think the OP is overestimating the impact of botnets though. Like another guy said, the botnet people probably have more profitable things to do with their time. Paypal scams and nigerian spam probably nets them a lot of money. Let's say the have a 100,000 botnet and each system averages ~3mhash/s. That's 300ghash/s. That's a lot of power. Some things to take into account though... Botnet PCs aren't always on are they? Surely some users must turn off their PC. And to think botnets will be accessing GPU miners is a little far fetched I would say. Think of the type of user and the type of computer that is in botnets, probably not too many powerful GPU's out there. Plus it would be a pain to configure each and every computer with the proper drivers and mining configuration for different GPUs, whereas a CPU client could be put onto every system easily.
I have no real idea which would be more profitable, typical botnet business or bitcoin mining, but my guess is that their typical business nets them more money.