Not to the degree that we do in modern society, but the Phoenicians did differentiate between the types of fish they ate. You can tell by looking at some ancient Phoenician relics which depict different types of fish being shown with different symbols. They also kept the bones of the northern blue-fin tuna (giant tuna), which also suggested they could distinguish between types of fish.

There aren't very many really top-notch answers so far on this thread, so I'll help out a bit. The answer is absolutely, completely, hell yes.

We can start with the Sumerians. Sumerian texts have quite a large number of nouns that are clearly different kinds of animals and wildlife. It seems that the vast majority of these nouns describe different fish found in the Tigris and Euphrates rivers. Unfortunately, the Sumerians were not particularly descriptive of what the hell they were talking about, and most of the actual species are unknown.

Let's move on to the Egyptians. The Egyptians did an awful lot of observation of wildlife, and in particular they have an enormous number of words describing very specific varieties of birds and fish. It's often been remarked that the Egyptians must have spent an enormous amount of time observing and cataloging this information. In fact, this effort must have been done almost entirely during the Pre-Dynastic Period, since even in the earliest texts there's a great deal of specificity in the words and signs depicting various birds. Because the Egyptians went to such great lengths to accurately depict the various animal-signs it's actually possible to determine what exactly they were talking about. Where it's somewhat uncertain we can often figure it out because the Egyptians are quite fond of describing in great detail the animals that they are talking about in their texts. There's a list somewhere (I'm not an Egyptologist but I'd imagine Gardiner has a catalog) of all the different kinds of wildlife in Egyptian texts. Some (actually a fair number) can't be identified, either because they're extinct or (more commonly) we just aren't really sure what that thing is.

Ok, so in Classical Greek we find a fair number of words for different kinds of fish. It's often noted that Homer never makes any mention of the consumption of fish in the Iliad and only mentions fish being consumed in the Odyssey a couple of times, always as a last resort to beat off hunger. There are many reasons and theories for this, but what is clear is that it doesn't necessarily depict the realities of a Dark Age or Archaic Period Greek diet, even for a noble. In the Classical Period the Greeks have a fair number of words for different kinds of seafood. Interestingly, words for squid, octopus, and various kinds of shellfish vastly outnumber words for actual fish (except for eels. The Athenians loved eels). Taken together with Homer's evidence it's been suggested that fish was a part of the diet of low-class individuals, something that was cheap and usually avoided by people who could afford it. There's some textual evidence to suggest this, and it's certainly true that fish was cheap and the most important part of the diet of the urban poor in both Athens and later at Rome, but that's not the whole story. The Athenians in particular seem to have had something of a taste for fish--Aristophanes has several jokes mentioning various kinds of delicacies (including those Boeotian eels that the Athenians were nuts about) and we have other references. So we're not entirely sure how they saw this

Now, Latin doesn't have many words for specific kinds of fish, usually just using the Greek word. But the mangum opus of all texts on fish is a work from the Second Sophistic by Oppian, an epic poem pretty much entirely about fish. To be honest, it's probably the most boring work of literature in the history of the universe (even though someone in my department is crazy about it) but it does preserve an enormous number of words describing different kinds of fish. The great thing about Oppian is that the vast majority of his fish are clearly identifiable, since he spends lines and lines describing them in absurdly specific detail. Another thing is that in many cases Oppian will remark on how the fish in question is viewed by its consumers, which helps us understand which fish were considered good and which were bad. The thing is, however, that Oppian's work is during the Roman Period, which makes it difficult to figure out when what he's saying is restricted only to that period or whether it's an opinion that was also held true during the Classical Period. If you'd like to know more about that pick up a copy of Oppian's works (the Loeb Library has them) but I really have to warn you--they're dull as paint.

In NZ, there are Maori names for all of the common fish & other underwater critter species, so I'd say that that stone-age culture differentiated between different types of fish pre-European contact.

I'm guessing you're talking about more complicated trade economies though, because it seems obvious to me that the people actually involved in catching and processing the fish would distinguish between different varieties.

edit: spelling.

Has any other seafood experienced a major swing like this?

I were once taught in history class that purpura (Tyrian purple) was extracted by sea snails in pre-Roman times (like around 800bc?) by Phoenicians to color the high class' cloth purple. Our teacher claimed that having access to these sea snails were a source of create wealth for them, and I'd imagine that they're not worth much now, or at least less than they used to.

Anyway It's not exactly what you're looking for, and will probably get deleted, but maybe someone can elaborate with a proper source.