From my summary reading around the internet, there seems to be a preference for meats as a protein source (e.g., see this SO answer). My question is, is there a difference—with respect to bodybuilding—between the protein from meat/fish/poultry, that from milk/cheese/yogurt, and that from peanut butter/nuts?
To best understand the answer to this, you have to understand that our bodies need Essential Amino Acids. In short any nutrient that is essential is one that your body cannot create on its own. There are a number of vitamins and minerals that fall in this category, but that is a topic for another question. Amino acids are the building blocks of proteins.
The primary reason that meats are the most preferred source of protein is because it's essential amino acid profile is the most complete. If you eat 50g of beef or chicken, you don't need to eat 50g of another source of protein to fill in the missing amino acids the same way you do for plant sourced proteins. Milk also has a fairly complete amino profile, better than peanuts, but not as complete as meat.
Plant born protein, whether from legumes, beans, etc. has a very limited protein profile. In order to get all your essential amino acids, you would have to pair one source of protein with a complementary source that provided the missing set of nutrients. This is why vegetarians may struggle to get the protein they need--particularly if they are active. The one plant born source of protein that has the most complete amino acid profile would be soy. Problem is that soy is a phyto-estrogen, or essentially it behaves like the estrogen hormone in your body. Too much can throw off your normal hormone balance.
The real answer to your question is probably that the supposed superiority (for bodybuilding) of these types of protein sources has been determined by generations of bodybuilders using trial and error.
I think, as the rest of this answer is intended to illustrate, that the number of variables in evaluating all differences between even a few different protein sources requires context and complexity beyond the scope of an answer here. However, I'll try to enumerate some of the factors that you can use to try to delve deeper into a comparison between some very specific (and by specific, I mean to include cut of meat, method of preparation, and method of raising the animal) protein sources.
The pragmatic differences between these protein sources (and others as well, including powders and vegetable proteins) can be broken down into two major categories:
Amino Acid Profile
Different types of protein are absorbed and used in different ways. Animal sources are generally recommended firstly because they tend to have amino acids in a proportion closer to what the body needs than e.g. plant sources.
Secondly, different protein types are absorbed by the body at different rates. Right after a hard workout, it may be beneficial to get protein into use quickly. However, for anti-catabolic purposes, such as while sleeping, (more about this later) it's useful to keep an elevated blood amino acid level for a longer period. For example, whey protein (the second-most abundant protein in milk) has a blood amino acid peak time of about 1 hour, with a return to pre-meal levels in 3-4 hours. Casein (the most abundant protein in milk) has a blood level peak about 1-2 hours after consumption, but the blood level remains elevated for up to 7 hours. Source
Thirdly, different proteins can have different beneficial effects with regard to bodybuilding needs. Again comparing whey with casein, we see that whey has a greater anabolic (building muscle) effect, but casein is superior as an anti-catabolic (preventing muscle loss). [Boirie Y et al. Slow and fast dietary proteins differently modulate postprandial protein accretion. Proc Natl Acad Sci USA 1997 94:14930-14935.]
Finally, different proteins contain different amounts of the branched-chain amino acids (BCAAs), which are strongly anabolic. I'll refer you to this question for more on these.
There is no such thing, really, as just eating protein. Whether you eat a steak, fish, tofu, or whey powder, you are getting something else besides pure protein. Animal sources will always contain at least some amount of fat (and of course, not all fat is equal either). Plant sources and powders generally also have at least some trace carbohydrate (and powders can contain all sorts of fillers). In general, natural "whole" food proteins will contain more micronutrients (vitamins, etc.), which may aid absorption (not to mention general health). So the choice of protein source has to account for these extras that come with the protein.
It's worth noting as well that there is no such thing as just e.g. "chicken". The profile is altered per the part of the chicken - breast, thigh, etc., as well as whether the skin is eaten, and possibly even how the chicken was raised. If you'll allow me to include the egg as part of the chicken for the sake of a clear example, it's generally accepted that free-range chicken eggs (because of the bugs that are included in their diet) have a higher omega-3 content than conventional eggs. Conventional beef tends to carry antibiotics, hormones, and extra omega-6 (hence the recent movement in many health communities towards grass-fed beef).
These adjuncts can have effects beyond the protein source in isolation. For example, adjusting carbohydrates (whether part of the protein source or from the rest of the diet) can affect the rate of weight (both fat and lean) gain or loss.