What is the difference between a DVD recorder for a PC and a DVD recorder for use with a TV?

Well and of course the other difference as you kind of stated yourself is that one for a PC will burn music, pictures, data etc....pretty much anything you want...software or backups etc.  The ones that are standalone that you record TV off of does not do that obviously.  As far as I know there is not a DVD Recorder that will hook up to both at the moment and do both.  The other option to go is to get a DVD Burner in a PC and a TV Tuner card and you can burn off TV shows etc to the DVD burner.  But that is about as close as you will get.  Which could be nicer anyway in which you can edit/modify the TV shows before you burn them.  Saving space and giving you more room (editing the comercials etc)

I think the chief differences between a dvd recorder for a pc and a standalone dvd recorder is that for a pc, it consists of just a dvd writable drive, whereas for a standalone recorder, it includes drive, analog-digital converters, power supply, support electronics and firmware programming to enable simple pushbutton operation.  The pc drive has the advantage of maximum flexibility, because you can attach as many features and sources and software as you want; the standalone recorder has the advantage of a single box solution and easy to use interface.  Which one you choose depends on how suitable the solution is for your needs, because different people have different requirements, and standalone units will also be more expensive because they have more parts and added value.

A standalone recorder can work with your tv and your pc if your pc has a video output that the recorder can accept - typically component video, s-video or composite video, in addition to audio output.  This is usually less than optimal in quality, because it involves two conversions: digital to analog, across the wires, and analog to digital.  In a pc, the process can be all in the digital realm.

When purchasing a pc recorder, you want a drive that can record in the format that most standalone dvd players can handle, and that would be DVD-R.  The good thing is that many new drives will do this and other formats also, like DVD+R and DVD-RW.  You will also need an analog video capture card if you are recording vhs or 8mm tapes and a firewire card if recording dv camcorder output.  Add also video recording software, such as Pinnacle Studio, and you will be set.