Both of these types of training builds the back, but the question is what can you stick to, and what is aligned with your goals?
If you want crazy, huge, jacked back muscles like body builders, that is great. This post might not be for you. If you want to lift the most weight possible, power lifting might be good, but this post might not be for you. Although I do believe body weight training can take many people very far in building mass and strength, it is important to keep in mind and be clear of what exactly you want. With that said, I feel quite awesome building the size and strength of my back using Calisthenics. There are limits to this, however, which I am ok with. It is certainly not optimal if my goal was to try and Lat Pull 300 pounds. Personally, I want to be good at pulling my body weight with two hands and one hand. I want to be good at pulling high to my sternum, and waist, rock muscle ups, be killer at front and back levers, I want to climb trees like a monkey, and I want to learn to rock climb.
These are just a few of the things I want to accomplish, and at the end of the day, I love getting better at Calisthenics. I know it will not be optimal for my absolute strength even though I have increased my absolute strength with Calisthenics. There are lots of people stronger than me and that is great. As far as weights go, lat pull downs, bent over rows, and seated rows, and the many variations of these exercises will build the back. Deadlifts, specifically, a deadlift variation that fits your skeletal structure well so as to maintain good form and tension, bracing properly, and all that good stuff will be key to improving your back strength as well as many other muscle groups involved in deadlifts. There are so many more details about these exercises, but I will not touch on them. There are so many amazing experts today, that I will leave you to find the experts that resonate with you on these exercises. Since my goal is to specialize in body weight, it would make sense for you to get more info on these from someone who specializes in these weighted exercises.
Anyway, pull ups are a great way to build your back strength. I have a pull up module on this training blog and I have a few videos on YouTube that provide some strategies to help your pull ups. Starting with a flex arm hang can help, as well as half range, and negative rep pull ups will help a lot. Negative reps is basically an eccentric overload, meaning you put a lot of resistance on lengthening the muscles. Basically you stand on a chair, or just jump up to the top and come down with as much control as you can. Over time you want to go slower and increase your time under tension. If this is even too hard, then it is good to get into a horizontal position, back facing the ground. Get a fixed bar or TRX and do your best to pull yourself to your chest. Bring your legs in as much as possible so you can safely perform the exercise. Your glutes should be contracted, core as well, shoulders down and stable. I like to say stiff as a board, lite as as feather. Create tension and direct it into your back primarily, and then arms. This is a great beginning to building your back strength.
As you get stronger, working on front and back lever progressions, and one handed pull up progressions will do wonders for your back strength. These exercises will further hit your biceps, lats, rhomboids, posterior deltoids, traps, and spinal erectors to varying degrees. These muscles are not targeted in all of these exercises: they all emphasize different areas of your back. For example, you will get more lats and biceps on one handed pull ups and more spinal erectors on back levers.
Ultimately, the choice is yours in what you do regarding your training. Whatever your choose, it is reasonable to start slow, perfect your form and technique, and increase the difficulty over time so as to try and reduce and prevent injury. And remember, small and consistent steps will serve you well throughout life!