The only problem you'd have transplanting your old disk to the newer computer might be the controller type. The newest computers have SATA drives, while older ones are PATA/EIDE. From the descriptions of your systems, they both would have EIDE drives, so all you need to do is remove the drive from the old computer and transplant it into the new one. There should be a spare connector for it if you have only one hard drive in the newer computer.
Pay attention to the jumper settings on the old disk. Depending on the manufacturer, you may have to change or remove a jumper to configure the disk you're transplanting as a slave drive. The settings will be labeled on the rear of the drive. In the picture above, the jumpers are located between the interface connector and the power connector, but it could be different on your drive.
Once the drive is installed, you should be able to boot up your computer, and Windows 2000 will see the new drive. However, it may not be automatically configured as drive E. If it isn't, go to Start, Control Panel, Administrative Tools, Computer Management, Disk Management. Find your new disk, right-click to bring up its context menu, and select Change Drive Letter and Paths... to configure the drive letter to what you want it to be.
You mentioned that you want to move some applications. This can be a little complicated. In order to successfully move an application from one computer to another, you need to preserve its run folder, application data, registry keys, and any extra DLLs it might have placed in the Windows folder when it was installed. This application might help:
http://www.snapfiles.com/get/applicationmover.html
Once you've successfully transferred the data from the old disk, you can reformat it. As you are moving it into a Windows 2000 computer, you have a couple of options. The disk will be formatted as FAT32 if it comes from a Windows 98 computer. You can elect to use the same format, or your can format it to NTFS. As it's a comparatively small disk, it won't matter which you choose. I'd keep it FAT32 just because it's a more portable format if you decide to move the disk once again to a removable enclosure.
