Here’s a sweet tip. If you want to get good at something, practice exactly what you want to get good at, it is the fastest and most efficient way to get better at that specified task as your body will eventually condition itself to become more efficient at said movement pattern (this is a simified way of explaining it). Other ancillary exercises can help, however not as much as the actual movements themselves. Think about people who back squat a ton, but don’t front squat, they are very efficient at that movement pattern, take them to do a front squat (a movement that involves similar muscles but requires far more stabization and is a different movement pattern) they may not even be able to do the bar. Similar to chin ups, some people can do like 20 chin ups but can’t even do 5 wide grip pull-ups. So the sweet tip is, do more pull-ups.
As to lat pulldowns, lat pulldowns allow you to use a lighter weight and focus more on the muscles contracting, to get a similar response from pull-ups you need to be strong enough to do them very slowly, focusing on the eccentric part of the movement. The forearms burn more with pull-ups typically because you are moving more weight. When I do pull-ups I’m lifting all my bodyweight, when I do lat pulldowns I use like 110-120 lbs and go really slowly, focusing on my lats contracting. Lat pulldowns are probably a better builder in terms of hypertrophy (increased TUT, more reps etc.) however pull-ups are probably better for overall athleticism, therefore a good approach is to do both (just don’t do any of that kipping pull-up stuff, in my opinion it’s like asking for shoulder impingement)