Weight training is often wrongly associated with getting "bulky," which deters some women from heading to the gym. That said, it's been proven not too, and instead improves your bone density, muscle mass, metabolism and more.
Training your shoulders specifically is key to good mobility - just think about how often you use your shoulders daily and you'll know that being to move with ease is important.
Lastly and as above, research from the British Journal of Sports Medicine showed that shoulder-specific strength training can also help with reduce neck and shoulder pain.