Softmod - Xbox is stock until the softmod loads when the dashboard files on the hard drive are accessed, so nothing can be changed until then. If your hard drive fails, you are back to unmodded. Upgrading or changing hard drive requires extra effort and understanding of the process.
Hardmod - TSOP or modchip - Xbox is modded on power on, so you can alter or disable the 'flubber' animation, disable the DVD drive requirement, and utilize an unlocked hard drive. Changing your hard drive can be as simple as installing a new one and booting an installer DVD to help set it up.
Hardmod - TSOP - main downside vs modchip - If you make a mistake flashing it with a bad or incompatible bios, or it fails for some reason, it can be difficult to recover from. You'll probably need to install a modchip.
Hardmod - modchip - Not all, but some have multiple banks to allow selection of different bioses that you can boot from. Helpful if you make a mistake flashing a bios, or you want to boot a different one for some reason. Some modchips can connect to additional hardware like an LCD display.
Personally I like TSOP - simple and free, unless you have a v1.6, then I would prefer a modchip over a softmod. Softmods are fine though, if you are okay with the limitations - they work.