I have found that if you're a casual web/email forwarder/digital picture manager then yeah, an old computer is fine.
I tend to push all my computers to the limit and I am VERY impatient with all of them. If you're impatient, RAM matters a lot (contrary to the other person who suggested that processor matters) because more ram means less trips to the disk for swap space and the trip to disk and back is usually slower than the time it takes for a processor to mull over something and come back with something else.

Just to give you my perspective on this.

At work, I typically have 10 to 15 applications running at once, on two large monitors. My "desktop" is usually configured as 3,840 Pixels by 1200 Pixels for my main computer, plus I have a laptop, another desktop for certain secure applications and I also access an array of test systems remotely. So my perspective is from a position of big jobs need big hardware.

But it's also from a perspective about old systems that comes from having a lot of them around. For most people, older systems, and yes, it's older WINDOWS systems, get flakier and stranger over time, and the answer always seems to be re-install Windows, and the problem with that is it takes a week for things to stabilize after a re-install and unless you're like me - so obsessed with protecting my data from computers - you're going to have to rebuild a lot of your file organization and storage, not to mention application settings and all that.

I make a living with my computer, and my time is worth money and faced with a week of lost productivity and hours of twiddling settings, I'll take the hit and dump a system that is exhibiting signs of brain rot, wipe the drive with an install of Ubuntu and give the darn thing away or move it into a non-important part of my life (like the workshop).