Ahhhhh...things are much clearer now! =-) I know that many people swear by the nylon wraps but the trainers at the place I train and groom horses don't allow them because they've had problems with them.
I don't know anything about the vet wrap so I can't advise you there...but I -can- tell you what I do and it works like a charm and the tail stays up for a good long time.
I start by slathering the tail in Mane and Tail conditioner and making sure it's combed nicely out with no knots. The Mane and Tail is great as a leave in conditioner and is really rather cheap compared to other things. I buy mine at Walgreens for like $5.50 for the big bottle. Then I start right below the dock and put in a rubber band to mark my starting place. I take some of the hay twine, making sure its long enough to braid into the tail and wrap around at the end, and wrap it around the rubber band and secure it behind the tail. Taking the two pieces of twine and putting them into the two outside sections, I simply braid down...making sure that the sections of braid stay even (borrowing from other pieces if I need to) until I get to the bottom and then secure the braid with another rubber band and wrapping the twine around that rubber band and securing it. (Still with me?)
I take the end of the tail (along with the twine obviously) and pull it through above where I started the braid. Usually this ends up being right next to the end of the dock on the right or left side. Depending on the length desired and length of tail, you can pull it through one or two times. I like to have the very end of the tail down on the bottom of the bundle to start the next part but that doesn't always happen. You just have to sorta go with the flow.
Next, I take the ends of the twine and weave them through the sections of braid on both sides..making sure to criss - cross them after they pop through to the front and wrap them around the back...weaving them through the braid sections again starting from the back a few inches up...repeating until I get to the top and then securing with a safety knot.
Last part...I take an old clean tube sock and cut it usually in half (generally only use the bottom portion but it depends on the length of the sock)...and then cut the top of the section I'm going to use crossways, leaving four strips to secure the sock. I slip the sock over the bundle and make sure that the end of the bundle is allllll the way in the bottom of the sock (cause I really hate when the sock looks like it's slipping off). Then I secure the sock by taking two of the sections and tying them together above the braid (about the same place you pulled the braid through.) Then repeat on the other side. I replace the braids every month and have never had an issue with them falling out or anything. In fact, they could probably go a lot longer then that with no issues.
I know my descriptions are a bit confusing so please don't hesitate if you have questions! =-)