I'm a big fan of weight training for several reasons. First of all, since it's the most pure form of muscle building, it allows you to specialize whichever muscles you want to build. If you're a runner, for example, you may want to do weight training so that your legs are not disproportionately muscular compared to your torso.
Also, while muscle strength isn't as beneficial to your overall health as heart strength, it helps you with all kinds of activities. I can't tell you how many times the strength I have acquired through weight training has saved me and others a great deal of time while moving furniture, renovating, etc. And you never know when you'll need to use your strength in emergency situations.
Finally, weight training is one of the most effective ways to look fit. Having big muscles does more than just give you a good image; it also helps you maintain that image because more muscle mass equals a higher metabolism. So if you lack the time to get regular, well-rounded workouts, you may want to focus on weight training.
When using weights, be aware of a few things. Weight training leads to all kinds of injuries if done incorrectly, so don't push yourself. For your safety, stretch before lifting weights, and keep in mind that the larger your muscles, the less flexible they become. Avoid exercises that stress your joints; it's not worth giving yourself life-long joint problems to get muscles that will atrophy over time. And don't overwork your muscles because if you do, they may not be able to repair themselves fast enough, and they may shrink; this is especially true if you don't get enough protein in your diet.