When a human is turned into a vampire, they gain many demon abilities that make them incredibly powerful. Most of these abilities are best used to hunt and kill prey.
The biggest difference is the fact a vampire gains extreme strength, and has much agility and reflexes. This is more than a match for almost every human alive, and serves the vampire well for hunting and feeding. Of course, like humans, some vampires are just naturally stronger then others. Also, if a human who is strong is turned into a vampire, then that human strength is added to the vampire strength, creating a very powerful vampire. This is why many vampire leaders will sire huge men; they make incredible bodyguards even against a Slayer.
Other enhancements include things like smell and hearing. A vampire can smell blood and other things, and has excellent hearing. Angel has even been shown to be able to tell if a bloodied crime scene contains blood from different people. This might be a skill Angel has picked up because of his enhanced senses, or maybe itís something all vampires can do.
The most noticeable difference is that vampires have the ability to change their face into a "vampire face" this face is vaguely cat like. When in this face, vampires also have cat like eyes, and fangs. Since a vampire is a hybrid between the vantal and human, the human face and the vampire face are both the "normal" face of a vampire. One is as natural as the other. However, as a vampire grows older, its demon side becomes more and more powerful, and eventually, the vampire will have a permanent demon face... examples of this are The Master and Kakistos. The face, oddly, will not totally resemble a vantal, this is odd because the demon side of a vampire is a vantal. However, there are similarities. For example, the teeth of The Master are a very similar design to the teeth of the vantal. One reason might be due to the fact the vampire is not a pure demon, this is possibly why certain vampires will develop into different types of demons, even if their demon blood is vantal. Another interesting thing is that most vampires, when angered, or injured, will uncontrollably change to their vampire face. This is probably because the demon side is naturally suited to violence, and will be dominate in such things.
Vampires do not age, their bodies are, for the most part, just reanimated preserved corpses, and so they, through supernatural means, stay the same forever. There are some exceptions, for example, vampires still appear to grow hair... though perhaps at a much-reduced rate. The show has never been clear on this, but vampires have been shown with different hair lengths at different times.
Because of this preservation, a vampire can suffer terrible injuries and heal from them easily. Since they can only be killed by a few select things, they can suffer injuries a human could not heal from, like a broken spine. Gunshots, swords, and any injuries caused by weapons that aren't wood can't kill a vampire, only cause pain. Things like cancer and other human illnesses would not affect a vampire. Certain vampire poisons and magic do exist though, which will permanently hurt, or kill a vampire. It's also been shown that if a human who needs glasses is turned into a vampire, they will continue to need the glasses. Poor eyesight is more of a "physical" problem: eyes will remain the same when someone is dead, where a disease would die with person. This is probably the reason.
Other example of the supernatural preservation is that vampires don't need to take in oxygen to live. They can, however, force air in and out of their lungs, which allows them to do things like smoke, or perhaps move cool air into their chest if they get too warm. There is much confusion as to why Angel couldn't perform CPR on Buffy. Perhaps he was worried about what he might breathe into her, as his lungs don't filter air anymore.
Vampires do not have a beating heart, and yet they still, through some sort of supernatural means seem to have a blood flow. Without a blood flow, a vampire could not bleed, or react to drugs, which they clearly do. This is a big mystery but the answer is that it does happen, for whatever reasons. Lots of things can be confusing when dealing with a dead body that is being kept alive because of demon properties, so it's best to not think much on how these things are possible, just that they are.
The last important thing to note is that with age, a vampire's powers will increase and they might even gain new abilities. This of course, is another study, and can be viewed here.