They both translate to "kind, type, category". They share one Hanja (유/류). It seems that 종류 is used more with real objects and 유형 with abstract concepts. Is that right?
유형 is more like "Shape, Characteristic", while 종류 has emphasis on "Sort"
Let me show some examples.
The protagonist of the drama is a typical type of father.
The example can be represented as "The protagonist of the drama has typical characteristics of father". Therefore, it is more appropriate to use 유형 in this case.
There are various types of goods in department stores.
In this example, it is more related to sort (group) , not their characteristics. Of course, they are sorted(grouped) by some characteristics or criteria, but more emphasis on sorting itself. Therefore, 종류 is more appropriate.
Types of mammals
Rabbit, elephant, lion, tiger, etc.. (a group) - 종류
Types of people who are not helpful to your life
People who are selfish, People who are mean, People who always lie, etc.. (a characteristic) - 유형