It is difficult to tell which one of them is better than the other, since calisthenics and weight lifting, they both equally depends on limited source which is the time.
Commonly, weight lifting, people do at the gym using heavy weight machine has great advantage on lifting heavy weight efficiently through close the gap between weight and center of bodyweight. Also weight lifting can gradually increase the muscle mass by controlling the weight of workout machine from light-weight to heavy-weight as muscle gets stronger.
On the other hand, unlike the weight lifting, calisthenics requires more of isometric contraction. Compared to other form of exercises it is very important to have healthy joint, since it uses bodyweight as its resistance to push it away from the center of gravity. It is better at aspect of functional movement but it is less efficient in increasing the muscle mass and get bigger compared to weight lifting.
When time is given equally, doing calisthenics over weight lifting to increase muscle mass is somewhat inefficient.
However, If you choose to do the calisthenics, it will give you a great advantage on the functional movement, so if the main purpose of exercise is to improve functional ability, then the bodyweight exercise is the answer.
Also it is easier to learn the fundamental movement of exercises. Let’s say it is your first time to do the squat, and if you are person who have no experience on lifting weight as well as the squat, then it would be difficult to do the squat while lifting light or heavy weight with proper form which is the key to building muscles and strength. However if you do the squat just with your body weight, you will be able to learn and focus more on which muscle should be keep activated during the exercise without the risk of injury.
Weight training and body weight exercise they both can give you a improvement of muscle mass, and functional ability, but if you want to improve your health, agility, power, flexibility, balance, and improve over all athletic abilities, then you should be thinking of starting a calisthenics.