Just as the names imply, animal protein is protein that comes from animals and plant protein is protein that comes from plants. Protein is protein is protein, though, right? Not really.
From an evolutionary perspective, humans are designed to eat animals. Whether we want to be or not, we are part of the food chain. In fact, there is no culture (during homo sapien times) that does not eat some form of animal product. The consumption of animal products is what helped our brains to evolve into what they are today.
Animal proteins contain all amino acids. This allows us to utilize it efficiently and effectively. Plant proteins do not contain all amino acids making it harder to utilize the protein. In fact, it has been found that people who consume only plant protein will actually require more to meet adequate protein levels.
Plant proteins are significantly higher in carbohydrates and lower in fat when compared to animal proteins. Below is a comparison chart of commonly consumed proteins according to Nutritionix.
Personally, I like to keep my insulin levels in check which means I want to keep my carbohydrates lower than my protein and fat. I also want to make sure I am getting adequate iron, choline, Vitamins D and B12 which are all only available from animal sources. What about the other vitamins and minerals, you ask? I'll get those from all of the veggies and some nuts and seeds. Maybe you're worried about saturated fat and cholesterol? I'm not. I'm more worried about sugar and inflammation. And according to this study, "We found no evidence that following a vegetarian diet, semi-vegetarian diet or a pesco-vegetarian diet has an independent protective effect on all-cause mortality."