For basics, not all proteins are the same, and this is an important thing to remember...
Also remember that correlation does not equate to causation... In this case just because you see a difference in your birds when switching from a plant to animal or animal to plant protein based food, it doesn't necessary mean the protein source was the reason.. There are great vegetable based feeds and horrible animal based protein feeds and vice versa, its not always about the protein source it likely a combination of many factors...
With the right combination of plant proteins and a few supplements (generally specific amino acids) omnivorous can be fed a balanced 'vegetarian' diet and do just fine... But, on the other end it's generally easier to formulate a balanced diet for omnivorous if you include animal proteins and plant proteins...
Most commercial chicken feed has plant proteins for three reasons...
1. There was and still are some concerns that 'disease' might be transferred by animal proteins in feeds (this has been proven to be false in most cases but still the concerns lingers)... For this reason some areas have outlawed and prohibited animal proteins in livestock feeds at times, thus as a proactive measure many feed companies switched over to all vegetarian feeds so they didn't have to worry about losing a market where animal proteins were outlawed...
2. Plant proteins and the necessary supplements in many regions are cheaper then animal proteins, so when profits are the goal the feed companies go to vegitarian diets, and the reverse might hold true in regions where cheap animal proteins are available....
3. Like many 'eating' trends, alternative diets like vegetarian are trending right now so some companies push that market...
I personally believe that a varied diet that best mimics what the animal would consume in nature is best, and with that I believe that omnivorous should have both animal and plant proteins in their diets and I strive to achieve a mix of both for my flock... This is why I rotate through multiple brands of poultry and even hog feeds containing both plant and animal proteins and also heavily supplement with fresh foods...
In the end there is no one right or wrong way, most commercial feeds be plant or animal protein are formulated to provide what chickens need...