Against Weight Training:

1) The haters of weight training say that you will indeed get bigger muscles lifting weights but you will also become slow and egotistical. You will become a hypertrophic, protein-dependent, muscle bound ape with little aerobic capacity.

2) Unfortunately, gyms are full of gym rats that use up their time with only cosmetic exercises. After they pump up their biceps and chest and put a shirt on, one would think they were smuggling four cantaloupes. They end up with a minuscule amount of real world functional strength.

3) Members of the bodyweight/calisthenic cult may tell you that many weight exercises are unstable because many are classified as open kinetic chain (OKC) exercises. The more weight you put on the barbell, the more prone you will be to injury.

Conclusion:

As was said at the beginning, the two systems of exercise are not two different belief systems nor are they two things of the same kind. One is like apples, the other, oranges. It's good to be adept at moving and pushing around your own body weight as it is also cool to be able to move around heavy things. Although I framed this article as a type of debate, there is, in actuality, no real substantive debate between weight training and bodyweight exercises. You should think in terms of specificity and do both for what you need to do.