I have a bootable USB drive w/Ubuntu 10.04. I use it to learn Ubuntu and to diagnose, repair, or recover other people's computers and files. The problem comes when I constantly have to keep re-downloading the tools I need. Is there a way to keep the tools I downloaded and keep Ubuntu they way I customized it (i.e. wallpaper, graphic options, icons on desktop, etc.) without installing it on the PC? I want to be able to keep everything on my USB drive.
If you use the USB Disk Creator (available from the Live CD I think, see here amongst many other places for more detail) you have the option of reserving space on the USB stick for holding updates. If you create a USB stick this way then changes, including updated and added software, should be preserved between boots.
Make sure you reserve plenty of space for such updates: you will no doubt need more than you think eventually as over time more and more packages need to be updated/added and any log files that are updated over time will get stored there to. If you only have a 1Gb USB stick, I suggest reserving all that is left after the system block is allocated for updates - I would recommend a 2Gb stick or larger though to allow more space.
It is possible to install a "proper" system to a USB drive too, though using the tool mentioned above is probably much simpler. The tool creates a live-CD like system plus a shadow file for updates, which is less efficient in some ways, but should work perfectly well for the use you describe.
If you have Ubuntu installed on the USB drive, I don't see why you'd have to re-download/reinstall software. Please provide more details.
But, if you're just using the USB to store a live cd, I can see why you'd have this problem. If this is the case, I suggest you install ubuntu on the USB drive.