In addition to keeping a record of what you installed in your user CP for future reference, it lets the author know that you appreciate his hard work enough to take the time to click the install button.
I have one hack released with only 4 installs, one of them my own, and all three times I've seen that number go up, it's made my day. I can't speak for any other coders on here, but it's very rewarding to know that someone else took the time to use your modification.
My .2c. Always click install if you're using the mod.