Lifting weights, or resistance training, has numerous benefits to the muscles and skeleton that are uniquely attributed to this form of training.
There are some cognitive benefits associated with exercise in general, but this FAQ entry will be more focused on what resistance training can give that other forms of exercise cannot.
Resistance training is a form of training where the muscles and skeleton are pit against a large force, either induced by external resistance (lifting weights) or by gravity (maximal jumping or sprinting). Resistance training tends to be focused on power, and tends to be anaerobic (intense) in nature.
Anything with maximal exertions can be considered resistance training. Things like Tennis and racquetball show some benefits as well due to some strides being full exertion, but weightlifting tends to have the most dramatic effects.
Benefits to Muscles
Most notably weight lifting, but all forms of resistance training, can increase muscle mass.
This can reduce the occurrence of sarcopenia (the age-related decline in muscle mass not associated with pro-inflammatory cytokines) when elderly, although all activity can reduce rates of sarcopenia, resistance training seems most effective.
Benefits to Bone
Exercise in general tends to be associated with better bone mineral density and/or bone width in athletes when compared to a non-athletic control group. Greater bone health and an exercise regimen are inversely associated with falls in the elderly, which suggests that exercise is a good preventative measure.
In older age, those who practice Sprinting have been shown to have better bone density and size relative to jogging and walking activities. Although beneficial bone adaptations seem to be better in the young, they can still occur even if one starts a physical exercise program later in life.