Protein is NOT protein.
Protein is made up of 21 different amino acids. Some of these amino acids your body can make from others, others cannot be made by your body and MUST be consumed as part of the diet. Those are called essential amino acids. Getting a protein with the proper balance of essential amino acids is not just important, it's essential.
Another difference is that some proteins are digestible and others aren't. Feathers, finger nails and fish scales are also protein, but they aren't digestible. Chicken feathers are a very common ingredient in pet foods, particularly in fish feed. It raises the amounts of protein in the feed at very little cost to the manufacturer. It does not however, add to the nutrition gained from the feed.
All proteins are NOT created alike.