Engaging the brain
Focusing multiple senses
One of the reasons that learning music has such affects more so than other activities is the fact that we are needing to fully engage three senses - hearing, touch and sight (sight less so as a musician becomes more proficient). In a study by Stanford which involved just listening to music (not even playing), "The research team showed that music engages the areas of the brain involved with paying attention, making predictions and updating the event in memory." (source)
Slows the aging process
I admit, I tricked you a little bit with this sub-heading, but what it has shown to do is diminish some of the negative effects of growing older, namely - hearing (also memory but we've covered that). Nina Kraus, director of the Auditory Neuroscience Laboratory at Northwestern University. It's not so much through better hearing per say, but better ability to discern what is being heard, quoting that "music fine-tunes the nervous system", “Playing music engages their ability to extract relevant patterns, including the sound of their own instrument, harmonies and rhythms. (source)
Extra tip: I'm telling this to myself as much as anyone else, but listen to new music, we all love listening to our favourites but immersing in new material challenges the brain to understand it. (source)
Also, my goal here was just to highlight a few key things but there are so many more. Here is a great list of examples backed up by reputable sources and covering a range of fields, not just cognitive benefits, but also social, health and educational.