Curing and Contracting Vampirism & Lycanthropy
As touched upon above, the Dawnguard expansion basically consists of two parts -- the first part that you do as a Vampire, and the second as a Werewolf. Becoming a werewolf (if you are not already one) is accomplished in one of two ways; either through the assistance of the Companions of the Circle, or by playing through the Dawnguard faction side of the expansion, as you are given the opportunity to become a werewolf as part of that.
Bear in mind that the Companions of the Circle will offer to turn you into a werewolf as part of their questline, and should you have either cured that condition or been turned into a vampire, Aela of the Companions will turn you back into a werewolf ONE TIME. After that, you are SOL, and that is something you need to bear in mind... Ideally then if you are already a werewolf you may want to go ahead and level the ability and gain all 11 werewolf perks before you even begin the proper Dawnguard story line just so that is out of the way.
If you are not a werewolf at the start of Dawnguard AND you have not completed the Companions quest line (Lycanthropy is contracted as part of the blood ritual that occurs during The Companions quest, The Silver Hand), you are better off simply playing through the expansion and being turned into one that way, as then you can complete everything and eventually do the Companions quests as well, which will leave you the option later of having Aela turn you back should you need to... Being turned back into a werewolf WILL cure vampirism.
To recap, becoming a Werewolf is only accomplished via the Circle of Companions in Whiterun or through the Dawnguard faction quests.
-- Becoming a Vampire --
At the cusp where you have to choose which faction you will join the Vampire Lord will turn you into a vampire if you choose his faction. Otherwise when you battle with vampires there is a very high chance that you will be infected by Sanguinare Vampiris, the disease that will eventually turn you into a full-blown vampire (but not a Vampire Lord).
Basically when you are in combat with a vampire, every time the vampire hits you with a spell OR a weapon, there is a 10% chance of you contracting this disease. This disease progresses to full Vampirism after 72 in-game hours (three days), but during this time you will receive notifications by the game that you have this disase in the form of hints about feeling strange when the sun goes down, or feeling thirsty when the sun goes down...
It is important to note that it is impossible to become a Vampire while you are already a werewolf, so if you are a werewolf you do not need to worry about combat with vampires infecting you. You also do not need to worry about using Cure Disease potions in that case as well.
In the process of writing this guide I was very active on the Dawnguard chat boards discussing different aspects of the game and expansion with other players, and one of the questions that cropped up quite a bit was that if being a vampire AND a werewolf at the same time... The reality is that you cannot be both at once time from an infected point of view, but you CAN be both a vampire and a werewolf if you are a full-blown vampire and you use the Ring of Hircine -- since the ring will temporarily turn you into a werewolf even if you are not one, because that is its special power. That being the case, if you are a vampire and you use the ring you essentially can use Beast Form while being a Vampire, which sort of makes you a hybrid of both beasts...
-- Curing Vampirism --
Vampirism can be contracted and cured an unlimited number of times -- but the cures are not what you would call instant or easy... Vampirism has four unique stages of severity, each of which is pregressed to the longer you go without drinking blood. Drinking blood will revert you to the next lowest of these stages, and if you do not want to get negative notoriety or bounty, your best tactic is to feed on sleeping humans. The reason you want to do this is simply to avoid Stage Four of the affliction, because the lower the stage the less likely your condition is to be detected by humans, but if you are Stage Four it is a certainty that you will be identified as a vampire and attacked on sight.
Method 1: If you are already a member of the Circle you can speak with Aela to have her change you directly into a Werewolf, which will instantly clear you of your Vampire affliction.
Method 2: Use the Shrine to Talos outside Whiterun or Dragonsreach; activating the shrine will cure you of all diseases.
Method 3: Complete the Rise of Dawn Quest
-- Speak with any Inn Keeper or Bartender about Rumors -- they will tell you about a man named Fallon in Morthal who is studying vampires;
-- Go to Morthal (East of Fort Snowhawk) and speak with Fallon and he will tell you about the cure for Vampirism;
-- Fallon instructs you to obtain a Black Soul Gem (the Black Star will not work for this) -- if you do not already have one, he will sell you one, but either way you need to use Soul Trap to fill that Black Soul Gem in order to advance this quest so do that next by killing a human and casting Soul Trap or using a Soul Trap weapon to kill one;
-- Take your filled Black Soul Gem and return to Fallon and talk to him to complete the ritual. Voila! You are!
So basically the one difference between werewolves and vampires is that it is a lot easier to become a vampire, and a lot easier to cure (remove) that affliction.
-- Curing Lycanthropy --
The condition of being a werewolf was, up until the release of the Dawnguard expansion, largely a function of the quest system of the Circle. Once cured via the Companions quest series, until the release of Dawnguard the only way to get it again was the one-time re-infection that Aela would do...
At the end of the Companions quest line the player can get themselves cured by completing the quest -- but that cure requires the severed heads of two Glenmoril Witches. With the head in hand, the player visits Ysgramor's Tomb, where they draw the spirit of the wolf from their body and defeat it.
Lycanthropy is overwritten by the Vampire Lord when you choose to join the Vampire Lord faction, so technically that is a cure that the player can undertake when they do not have access to the witch head -- by basically getting turned into a vampire and then curing the vampire affliction.