My old HP computer is 8 years old, Pent 4 and XP. The IBM is 4 years old and Core 2 Dual but needs an O/S.
Linux can't work miracles, but I'm typing this on a 1.5 Ghz Celeron with 512 Megs of RAM and a dog slow drive... With a full KDE desktop.
-The biggest 'boost' for Linux is that you don't have to run a heavy handed Virus Scanner and Firewall. Those are the two things that eat a lot of memory and CPU.
-Whatever Linux you choose to install, try to install as few daemons / servers as possible, and disable the ones you don't need. Google and your distro documentation is your friend. You really don't need to be running Apache.
-When setting up Linux, when you set up the drive, look for a way to set the 'noatime' option for a partition. (This basically disables marking the last time a file was accessed. With atime, you turn every read into a read and write which is much slower. It's been called one of the biggest Unix design mistakes of all time.)
-You'll likely be running GNOME or KDE as your desktop. Look for ways to turn of animations, window effects, and file indexers. (Once again google if you have to.)
-Instead of GNOME or KDE you could choose a lighter desktop environment like LXDE.
The HP computer will probably need a fair bit of the above, the core duo laptop will probably be fine with less of the above since it doesn't have to run bloaty Windows, with a bloaty Firewall, and a bloaty Virus Scanner.
@Woodpecker-- Have you tried setting 'noatime' in your /etc/fstab for your partitions? It really is a massive (double digit) performance gain. There are only a handful of pieces of software that rely on the atime (none of which you probably run). You could also try relatime which is a less strict atime. On slower disks, like a lot of laptops, you'll be shocked at the performance boost.