I'll add my two cents here as I recently had a similar issue: motherboard finally gave up the ghost after five years of abuse and my one year old daughter banging on the on/off switch as a form of 'sparkly power light LED entertainment'. 
The hard drive managed to survive this and was okay. Rest of system didn't fare so well! Time for a complete gut-out of box and upgrade! Happy days. 
So picked up new motherboard, CPU, 2Gig of ram, two 250Gig SATA(II) hard drives, video card, new power supply unit. Ripped out guts and packed in the new gear. Hooked up the two SATA drives to motherboard and configured in a RAID 1 set-up (mirroring). In layman's terms, RAID 1 makes a complete back-up of one hard drive to another. (I used to like nothing better than fiddling with wiping-out hard drives and reinstalls for days on end. Don't really have the time these days, so RAID 1 is now my current preference). You can also set up the two hard drives in a RAID 0 set-up, which effectively spreads the data across two disks. IE: Two 120 Gig hard drives would give you a total of 240 Gig.
There are other RAID set-up's, but they start to use more than two hard drives and get a bit complicated to explain with regards to parity error functionality and mirror / striping combinations. 
Took me a while to get video card working: Long story short, had to update motherboard BIOS, which helped with over current supply power settings to video card. Go figure.
Reinstalled all programs on new HDD's and system from scratch. As ColdSteelRain pointed out, simply transferring the programs from old to new can cause problems with motherboard conflicts, registry entries not matching up, etc.
Transferred all data from original HDD to new HDD's, and clean wiped old HDD. Will chuck old HDD into an external caddy at a later date to use as an external hard drive.
In regards to your OP, I would probably recommend using the 'old' hard drive as an external unit, sticking it in a caddy and using it as an external back-up, or portable device for extra storage, etc. RAID set-ups work at their best efficiency when the two internal hard drives are identical.
Or, as per other posts, you could have it in system as a separate drive internally on a different letter IE: D:, but in that case, why not have it externally? Gives you the ability to cart around, hook up to work computer for file transferring, etc.
If internally, will 'old' HDD always be on? Yup, as long as the box is fired up and receiving juice, it sure will! (Unless the internal HDD doesn't have the internal power attached! 
) Externally, you can turn the caddy off and unplug HDD whenever you want. Caddy will have to accomodate the type of connection 'old' hard drive is IE: IDE or SATA, and then I think externally, they come with a number of different style of connections form caddy to computer. (USB, firewire, wireless, etc). Ask your tech guy for a bit of advise on this, as he knows your system better than we do!
