For all intents and purposes, a complete protein is a complete protein. HOWEVER, a complete protein is defined one that contains sufficient levels of all essential amino acids (aka ones we cannot synthesize naturally). Some proteins are higher in certain amino acids than others, which can have desirable (or undesirable) effects for the body. Whey protein, for example, is naturally high in BCAA type amino acids whereas soy protein (say, tofu) is not. If you know anything about BCAA's, you can see just how different a protein source can be from others, even if it has the same macros. It goes beyond rates of absorption and digestion.
I personally include all the protein types I possibly can. The only time I intake whey protein (mainly because I can't digest it, but even if I could) would be post workout, or pre-workout if I'm doing fasted-type cardio (precisely for it's natural 5g of BCAA's per 100kcal). I ingest a daily diet that contains: Eggs (water-soluble albumin protein), pork, beef or tuna, chicken, salmon and TVP (a soy protein concentrate). TVP is great if you're looking for both fiber and protein, as it's high in both, but it's a bit hard on the gut.
Any other time of the day except PWO, you should try and consume your protein from whole foods. If you had to choose between 200g of protein from food sources, or 300g of protein mainly from shake sources, I would choose the food. Every. Single. Time. There's more to food than protein and Macros. There are literally thousands of chemicals, isomers and variants of these chemicals just waiting to be eaten. The world is your oyster, or something, so go fucking eat it.