There are a lot of similarities between athletic training and musical practicing, however, I think it is a mistake to call them the same, at least for now. My reason is this. While both are inherently artistic, one is socially seen as sport and competition more than art. I feel that this is also creeping into the music world a little bit, especially during audition season. However, when both are considered and treated as an art form, they can be seen as equals in the results and the products. The misconceptions arise when we begin to compare the processes of both, the most common being the "no pain no gain" slogan.
Both require the increase of body efficiency while performing an action. Athletics usually deal with bigger, slow twitching muscles (body building, basketball, football), and music often deals with smaller, fast twitching muscles (piano, percussion, violin). The former deals more with the breaking down and rebuilding of muscles to gain strength or speed while the latter doesn't necessarily have to break anything down, but needs repetitions to become accustomed to the action being practiced, dealing more with the natural weight and resistance of the body or instrument. There is overlap in which muscles are worked out, running faster or lifting heavy cymbals, and knowing how they all contribute to the product will ensure that we don't over or under work any part. For these purposes, I'll take the cymbal example and go through how I dealt with navigating the weight of the instrument and how that resulted in changes to other instruments.
Cymbals are heavy and if you aren't used to "heavy lifting" there will be some soreness if practiced often or even too much. Even if you only do a little bit, your muscles may still be sore afterwards. In this case, the "pain" is actually natural soreness and will heal and grow stronger muscles that will make it easier for you to lift and play the cymbals. Because there is now new muscle tissue, it effects everything else that it is connected to. So when we go to play soft snare drum, it doesn't necessarily work the same as before because now there's more weight, muscle, behind the wrist. However, it might make playing tambourine rolls easier because of the extra weight and power/endurance in the upper arms. None of these side effects are bad, but should be observed very carefully to avoid diminishing returns.
Now, the issue comes when over or misuse of the body arises. Usually, the culprits are either a desire to be faster or louder (probably won't see many people tensing to play slow or soft, but it has happened). The misconception is to just start playing as fast/loud as we can as long as we can to get the "er" on the end of them. Actually, this is more a practice of endurance rather than gaining strength, power, speed or agility, and treating an endurance exercise as a strength one is what causes this over/misuse the fastest in my opinion. The other cause of pain and injury comes from the misuse of the parts that are doing the action. Let's take lifting a box for example. We all know we are supposed to lift from the knees along with everything else, but the main producers of the lift are the knees and legs. If we just lifted from just the arms or the back, we would over work them and risk damaging them. The same goes for when we practice or play. If we have to play loud, why just leave it to the fingers? They are small and weaker in comparison to the upper arm and shoulder. Why not put them in the mix?
As far as speed, we don't need to worry about gaining strength or power, but faster reflexes. The reflex that we have is a combination or the management of the natural weight of the appendage that's moving, the wrist, finger, etc, and how quickly they can twitch without "effort". Finding the balance between working with the natural weight and quickening the reflexes will result in faster speeds. Note, if you increase the muscle mass of the small muscles, they often will lose speed (every body is different but most often this is the case), which is why it's important to work mainly with the natural weight of the appendages you are using so they naturally grow and develop. Along with this, we do have to consider the instrument itself as well and how our bodies interact with it.
With the cymbals, the weight is something that we have to work with and get used to. It won't just get easier the next day, but takes time for the body to incorporate the new sensations and weights. Similarly, if we are playing scales on a clarinet, we would observe and figure out how we press the keys to avoid over or under pressing, freeing us to play faster and easier. Even if it's difficult to do at first, nothing should feel tense or stressful. Pressing the key should feel easy, the coordination will come with time and practice. The key is to not over try, and to be patient, easy does it. Know that the absence of pain actually means the presence and likelihood of gain in whatever you are doing.
Personally, I do consider and treat my practice in an athletic light, but I also consider it spiritual, philosophical and artistic, and I base everything I think or do during and after set aside practice times on them. I exercise based on the needs of the heaviest instruments, the loudest volumes, and the fastest speeds, no more than is necessary for the instrument. I'm always working on my agility, flexibility and reflexes, and anything else that can better the efficiency of my body in my work, and this is the motivation of most, if not all, athletes. Still, exercising is not necessary most cases, and you should base anything solely on what is needed to accomplish your goals, not on what other people do. If there is pain, not soreness from new muscle, something is wrong and you should stop immediately and examine what caused it and why.
Michael Jordan at first did not want to work out because he thought it would negatively effect his game, until he found a trainer that would help avoid any diminishing returns. Arnold Schwarzenegger viewed his body building as becoming more beautiful and artistic in his appearance. Janos Starker would swim daily to keep up his physical fitness to perform cello. All based their work on what was needed and avoided any unnecessary strain and injury. What's the need and how can you meet those needs? Be mindful of what you are doing and be careful when trying anything new. Stay healthy and you will prolong your career.