Welcome to BW and to the great world of Beagles. First, I would like to command you for inquiring about the breed of your choice before racing and purchasing the first cute little pup on your path.
Yes, there are difference between Beagles. Firstly, as you have mentioned, you have the 13" which is a dog measuring between 11" and 13" at the shoulder (meaning, from the ground to the top of his shoulder), and the 15" which is above 13" up to and including 15"
Then you have the "Show" beagle and the "working Beagle. The "Show" Beagle is a Beagle which has been bred, through the excercise of selection by his breeder, for the sole purpose of show. Along the years, people have develop a list of certain caracteristic which each dog belonging to a same breed should display. These caracteristics includes, hight, weight, colour and type of coat, angulation of the legs, neck, shape of the skull, the teeth... and also certain temperament traits such as "always on the move" or loyal, or aloof.... This list of traits are called the Breed standard. A Show dog is a dog who has been bred keeping in mind the standard, and in a dog show, the dog who is the closest to his breed standard, that is the most representative of his breed wins. Now purchasing a pup from a show line does not mean that the dog is of show quality. It only mean that he was bred for the sole purpose of show. There are no purfect dog, which means that no dog possess exactly all of the traits described in the breed standard. What makes a show dog is the balance of the faults. Certain defects are more faulted than others, and those who are more likely to prohibit the dog from being able to do the job he was originally bred for is generally more faulted. For exemple, an ear a little short in a Beagle is not as faulted as a Beagle with angulations too straight, which would stop him from moving fast and properly while hunting, even though the dog will never hunt.
The working Beagle is a dog who has been bred for the sole purpose of his work. That is, although his breeder will ensure to breed a Beagle with a Beagle, he will be looking at the performance of the dog rather than his look. And just as the breeder of a Show Beagle tries to breed a female with known and recognised qualities according to the breed standard, to a male who can either complement that female, the breeder of the Working Beagle will be searching for a great hunter, good strategist, fast runner, and with a clear voice, to use with his female the breeder believe to possess great working ability, thus getting good working pups. The result in these working lines are often that a Beagle will be over size, or with longer legs, or too fine of a skull, or all of the above. In the world of dogs, the discussion of the show dog vs. the working dog is endless and sometimes quite animated.
Now how do you know what your puppy will be like? The first thing is to TRUST the breeder. Go visit several and talk to them. Ask them what are the qualities and the faults of their line. Make sure the parents have been screened for genetic health defects, and ask to see the grand parents if you can. Pups usually look like their grand parents and not like their parents. Certainly, insist on seeing the Mom. The state and temperament of the Mom can tell you a lot on the way the pups were kept and the temperament a pup will have. It is often easier to get information from a breeder of show dogs than it is from one of working dogs, but that does not mean that you cannot get a good working pup. I always fine that the way the dogs relate to the breeder is very representative of the way he looks after them. A pup born and brought up inside the breeders home is usually a better pup than one raised in kennels as he will be more people oriented. Finaly, make sure to get a health guaranty, no less than one year, covering all hereditary deseases, and try NOT to bring puppy home before he is 12 weeks, 10 at the very minimum.
Well, it seems like as always, I went on and on and I could probably continue for hours... I hope I have answered your questions at least in part. Should you require any specific information, please do not hesitate to ask. There are always people hear with the knowledge you seek. And please, do come and visit from times to times. We would love to hear all about the beagle you will be getting, and to see pictures too. We just love pictures...