Here I use the 28A, 37A/38A, 36B and 38A/B Connstellations as an example because they are (still?) a popular and reasonably well known instrument. However, most of this applies equally to other Conn cornets and trumpets available in short cornet, long cornet and trumpet models.
Wouldn't it be nice to own, say, a Conn Connstellation trumpet? Not one of the new ones you see in the shops these days, but an original pre-1970 model. So you start looking around on the internet to see if you can find one somewhere. Perhaps remarkably, you find that they are actually not all that hard to find, not accounting for price and variations in condition. But you will also find that what from the pictures appears to be the same model of trumpet can sometimes be called a "28A Connstellation", "38A Connstellation" and at other times "38B Connstellation". What's the deal?
The short answer is, that the "38A" usually referred to is a cornet, and the 38B is a trumpet. More to the point, the "38A" usually found on the internet is a so-called "long cornet" which model number is actually the 28A as far as I can tell from the 1957 and 1959 Conn catalogs and a 1962 Conn ad. I have convincing proof that it wasn't renumbered to 38A at some later date, since the "38A" model number was being used by the Connstellation short model between 1961 and at least 1968. Hence the confusion, I suspect. From this point on the Connstellation Long cornet will be called the 28A. The Connstellation short model cornet is called the 37A (brass bell) and 38A (coprion bell). A "long cornet" is a cornet that is wrapped ("folded") like a trumpet, but it takes a cornet mouthpiece. The shaft of a cornet mouthpiece is slightly shorter and narrower than a trumpet mouthpiece, so a trumpet mouthpiece won't fit a long cornet and vice versa. Because a cornet mouthpiece is slightly shorter than a trumpet mouthpiece, the 28A will end up being about 3/8" shorter than the 38B. The instrument itself without the mouthpiece is the same length. Officially a cornet is a conical bore instrument (it's a cone from the mouthpiece to the bell) and a trumpet is cylindrical ("straight"). Nevertheless the 38B Connstellation (trumpet) is quite conical, however the 28A long cornet, as I understand it, has a cylindrical leadpipe covering an inner pipe which is more conical than a trumpet leadpipe.
A word about the 36B Lightweight Connstellation. What is a "lightweight" Connstellation? Essentially, a 36B Lightweight Connstellation is the same as a 38B except it has a bell diameter of 4 5/8" as opposed to the 5 1/8" of the 38B (and 28A, for that matter) and the bracing is somewhat lighter. Models 36B through (at least) 1962 don't have a first slide trigger. The 1966 Conn catalog shows the 36B with a trigger on first, so sometime between 1963 and 1965 it gained the trigger. Otherwise it is identical: nickel plated with brass trim, wide wrap, top spring valves, etc. This makes it somewhat difficult to distinguish a later model 36B from a 38B. The easiest way is to compare the main tuning slide brace. The other way is to measure the size of the bell (that would be a hard thing to do from a picture). The 36B is a little less heavy than the 38B because it has a smaller bell (less metal involved). I guesstimate the earlier version without the trigger at about 2 lbs. 10 oz (it feels about the same as my 6A Victor). The lighter weight and bracing makes the 36B noticeably easier to blow and "get moving" than the 38B but still quite a bit heavier than something like a 22B New York Symphony. The use of a smaller bell does make some difference in projection, but since the 38B project exceptionally well I certainly wouldn't want to say the 36B lacks in that respect. The color of the sound of the 36B is very similar to the 38B. The difference is more of a "feel" thing. Overall I find the sometimes looked down upon 36B a very enjoyable instrument to play, especially if the 38B is a bit heavy for you in any respect.
How does one tell the difference between a 28A long cornet and a 38B trumpet short of inserting a mouthpiece to see if it fits? Usually there will be a letter "A" or "B" stamped on the mouthpiece receiver to indicate this. Also, if you look along the leadpipe towards the main tuning slide, at one point the diameter of the leadpipe increases to accomodate the main tuning slide. On a trumpet this increase in diameter occurs immediately after the finger hook on the leadpipe. On a long cornet this increase in diameter occurs about halfway between the finger hook and the main tuning slide. See under 28A and 38B for examples of this.
However, there is also such a thing as the 37A/38A short model Connstellation cornet. Now this instrument is instantly recognizable as a cornet. It obviously is of the same basic design as a 28A long cornet or 38B trumpet, but it is a short model, around 17" whereas the 28A Long version is just over 21" long. It is nickel plated and has a first valve trigger just like the long version. Never having played one I can't tell you if it "feels" the same way when played. The bore apparently is a #2 1/2, or 0.485" which is a lot larger than the #1 or 0.433" bore of the 28A long cornet and 38B trumpet. This almost certainly makes a quite a bit of difference to the way it sounds and feels, but having never played one, I can't say for sure.
In general a cornet sounds a little "darker", not quite as "bright" as a trumpet, depending a great deal on the mouthpiece you use. A cornet doesn't sound quite as "penetrating" as a trumpet. One cornet in a group of trumpets might have a bit more trouble being heard because of this. This doesn't mean a cornet can't play just as loud as a trumpet. Volume isn't related to the "color" of the sound. The conductor might like the cornet sound better (I am speaking from personal experience here). Traditionally the sound of a cornet is a bit more "mellow" (more flugelhorn like), but if I am not mistaken in the past people tended to use deeper mouthpieces with a cornet than is common these days, although such mouthpieces can still be found if you know where to look. Vincent Bach trumpet and cornet mouthpieces differ only in the shank, and not in the cup. In my band we use cornets and trumpets interchangeably. But I digress. In my, admittedly limited, experience playing short and long cornets the long cornet sounds about halfway in between a short cornet and a trumpet with regard to how bright or dark it is. My experience in this stems from playing the 15B Director and 38B Connstellation trumpets, a 6A Victor long model cornet and a 77A Connquest short model cornet.
Perhaps the short cornet weighs less than the long cornet or trumpet? Well, no, not really. Both the 28A long cornet and 38B trumpet weigh in a 2 lbs. 12 oz. Although I have no data for the 37A/38A short model cornet, I don't expect it to weigh less than 2 lbs 10 oz., since the 14A and 18A Director and 77A Connquest (short model) cornets all weigh 2 lbs 10 oz. Quite possibly it is the same as the 28A and 38B. The main difference in perceived weight is the center of gravity of the instrument. A 38B is about an inch longer from valve casing to mouthpiece than a (more) "standard" trumpet, such as my 1964 Director 15B (I own both types). Although the 38B is probably only an ounce or two heavier than the Director, it feels quite a bit heavier because you hold it an inch further away from your body. The 28A long cornet is the same as the 38B trumpet in build, so I expect both to feel the same way. As for the 37A/38A short model cornet, since it is several inches shorter than the 28A or 38B it will feel a lot less heavy eventhough it is about the same weight, because it is held much closer to the body.
So what is the upshot of all this? A short cornet will be less heavy to hold than a long cornet or trumpet. The larger bore will definitely make a difference (correct me if I am wrong, but that would make it a bit more mellow sounding, and require a larger volume of air). A long cornet "feels" a bit different from a trumpet because of the differences in the shank of the mouthpiece and the leadpipe. Ideally I guess you would have to try all three to form an opinion.
For a discussion on how to judge the approximate date on a Connstellation, see under 38B Connstellation.
38B Connstellation
28A Long model Connstellation
37A/38A Short model Connstellation
36B Lightweight Connstellation