To understand the differences between these two compound words. you need to first understand the meanings to 加 and 长。 While it's simple for 加 which just means addition (1加1=2), there are fives meanings to 长：1, grow; 2, length; 3, long ; 4, leader; 5, First
- 长大=grow up，长高 grow taller, 长长 grow longer (zhang chang) same word sounds different.
- 长夜=long night；长城=great wall of China which means long wall;
- 班长=Leader to a class; 市长=leader to a city; 厕所长=leader to a bathroom；
- 长女=First born daughter to a family; 长老=oldest old people in a family;
增 simply means getting more;
so 增加 simply means getting more in addition.
but 增长 can mean a lot of things. Refer to "长" meaning from above.
Sometimes you can't use 增加， but you can use 增长, and visa versa.
For example. You have enlarged your penis. and you can say：我增长了我的阴茎。You can't use 增加 here， because you are not adding （加） something to your penis because you are only making it grow longer. 长 used here means 1, grow. But of course, if you had an accident and have part of your penis damaged and you have done a penis transplant, then you can use 我增加了我的阴茎。