A couple needs a photo gallery for their wedding, where everyone can browse and upload their own photos from the wedding. Beginner developer: I know of some great software we can use, give me a couple of days. Skilled developer: I'll write something up from scratch for you, it will be perfect, give me a couple of weeks. Wise developer: Let's just use a Flickr group, give me a couple of minutes.
"Reinventing the wheel" on the web means doing something from scratch vs. using a pre-existing solution, especially referring to solutions that already work particularly well. On the "reinventing" side, you benefit from complete control and learning from the process. On the other side, you benefit from speed, reliability, and familiarity. Also often at odds are time spent and cost.
This screencast takes a look at a few examples that have come up for me recently. The answer, I propose, is like most things in life: somewhere in the middle.