몸무게 and 체중 both mean body weight, and both are widely used. 몸무게 is a native Korean word, while 체중 is hanja(Chinese characters)-based: 체=몸=body, 중=무게=weight. 체중 can make other hanja words like 과체중(overweight), 저체중(underweight), 체중 감소(weight loss), and so on. 몸무게(를) 재다 and 체중(을) 달다 both mean measure/weigh body weight, but 달다 is more commonly used. 재다 means "take measurement (using an instrument)" of any kind including weight. 달다 specifically means "weigh". 체중이 늘다 is literally "the weight increases", and thus to gain weight. 살이 찌다 is literally "살(flesh)이 찌다(grow, accumulate)", so to put on weight or to become plump. 살이 찌다 is more widely used for putting on weight, but 체중이 늘다 is also common. These phrases usually refer to a relatively small change unless adverbs like 많이, 굉장히, 엄청, etc is used. When someone gets really heavy, we say 뚱뚱해지다 (to become fat, 뚱뚱하다=be fat/overweight).

thank you :)

I'm not Korean, just a learner. My understanding is that 몸무게 (몸 - body, 무게 - weight of anything) & 체중 = body weight, whilst 살 (찌다) is referring to putting on weight (살 = fat). 체중 늘다 means to increase weight, which could be a positive or negative thing, depending on your circumstance. 몸무게 재다 vs. 체중 달다 = both (i think) have similar nuance of weighing yourself, or someone else. I hope this helps.