I took up weight lifting in the fall of the 2004, and have been periodically keeping up with it rigorously ever since. My current pattern, which is improving, is to spend about 3-4 months lifting regularly alternated with of 2-3 months meaning to go to the gym without ever getting there.
I took up weight lifting primarily for physical fitness. Weight lifting not only increases strength and bone density, but a lifestyle which includes weight lifting can really help to keep a person youthful into their 50s and 60s. I want to lay a good foundation so that I'll be able to count on my body in the decades ahead. Especially since as a programmer, my natural tendency will be to waste away into a couch potato.
I'm also trying to use weight lifting to augment the aerobic exercise I get (see the cycling section) to lose weight (technically, to lose fat; weight lifting will add weight because muscle weighs more than fat). Weight lifting itself burns calories, but more importantly simply having muscle burns a lot of calories. Muscle is calorically expensive for the body to maintain, which is why the body likes to atrophy muscle when it can. Consistent weight lifting combined with regular aerobic exercise can produce much better results than aerobic exercise alone.
Strength training also has a short-term practical benefit: when you work out and get stronger, everything around you seems lighter. Carrying things is easier, and lifting things out of inconvenient places takes less work. Getting stronger simply makes the normal tasks of life less burdensome.
One of the other great benefits of weight training is that it stimulates the body to get faster at recuperation. When I first started lifting weights, I was fairly sore for the first few weeks. After a month or two, though, I was barely getting sore. These days, I almost never get sore after working out. And that means that I also never get sore even after a full night of dancing.
The above was written in 2008. Since then I've continued to lift weights, though I sometimes end up stopping for 6 months at a time. I still very much enjoy it and its benefits, just like most things it's hard to find the time to be really consistent.