You can probably use the files and settings transfer wizard in windows XP.

I'd like to take the hard drive from my existing PC and install it in a new PC I am building. I would like to use the hard drive as my primary drive (for OS and applications) on the new machine. I will have a different motherboard and chipset than on my existing PC. What problems will I encounter if I just pull out the drive from my old PC, connect it in the new PC, and boot up?

Depending on the chipsets of the 2 different motherboards, you can get anything from a working machine where all you have to do is install the drivers for the motherboard and any other new hardware, to a machine which simply will not boot. If you get the latter, you could try to reinstall windows from the CD using the repair option (not the one from the command prompt) and then install all drivers for the new motherboard, updates etc. But either way, there will be drivers etc left on the drive that shouldn't be there. If you uninstall all motherboard, sound and video drivers from the hard drive BEFORE you take it out of the old computer, you may increase your chances of getting a first scenario result.
But it's always better to do a clean install on a newly formatted drive.
Good luck.

If you intend to use your old hard disk to your new PC that is very possible you just have to make sure that your old hard drive remains the master and configure your new hard drive as a slave.

A different motherboard and chipset?.. it will not boot up, that's it. There is a fair to very good chance that you will have the wrong version of HAL, and certainly wrong drivers etc. A Windows Repair will not be a good option, either, because all that will do is take the old registry hive and copy it into %windir%\repair [and that will do you no good if you ever need it], rewrite some parts of it to form the new reg hive, and rebuild system files. Mostly, your apps and settings would be preserved.
Reinstall Windows.
Your old data files will not be overwritten if you do NOT format [but not doing a full format can give data recovery software a headache if in the future you should need to use it] ... You may lose email client files though. Somewhere during Setup make the choice to leave the current file system intact but let it delete the old Windows folder.
=Personally though, I would copy off the wanted data, downloaded application installation files etc to that new slave drive, build the sys, format and make a system partition about 8GB in size, install windows and make more partitions for data and apps. Copying out data will save you taking ownership of special windowss folders like My Docs. With a Repair chances are you will keep some of the old problems and instabilities that your old Windows was building up.

I've done this several times.
A repair install has a good chance of succeeding, but it's not the right way to do this.
All the reasons have already been pointed out.
If you install the old drive as a slave, you can just copy your data, but you'll need to reinstall all your programs.
Otherwise, make copies of your data, reinstall, then copy the data to the appropriate places. The F & s wizard is good in this scenario.

Here is my updated situation. I completed the new build, and, as recommended, by a couple of earlier responders, I installed a brand new copy of Win XP to a brand new hard drive. Now I would like to put my old hHD into the new build, transfer all my apps from the old HD to the new one (the new one is 10,0000 RPM - much faster than the old one), and use the old HD just for data. storage My question is, once I put the old HD into the new build, how do I go about (safely) getting rid of WinXP from the old HD? I don't want to have two versions of WinXP installed on the two HDs - that just feels like trouble. But if I run Windows installer to uninstall WinXP from the old HD, how can I be sure I don't accidentally blow out the brand new WinXP install on the new HD? (This may be a stupid question, but I don't want to make a painful mistake here.)

first ,the old apps if install on the old drive will not just transfer and work on the new drive ,transfer of you own personal data is about all you will get .
to format and remove the xp type this into run."compmgmt.msc" without the " and then go to disk management ,.

Did you finish your message? It seems to end mid-thought.