Immune to disease and poison, 25% more powerful illusion spells, 25% harder to detect while sneaking, 25-100% Cold resistance, depending on stage of vampirism (1 being the lowest resistance, 4 being the highest).
While in stage 4 vampirism humans attack you on sight; Health, Magicka, and Stamina do not regenerate during the day and their maximum values are reduced (stage 1 grants the lowest reduction, 4 grants the highest); 25-100% fire weakness, depending on stage of vampirism (1 being the lowest, 4 being the highest).
You also get a certain amount of special powers based on your current stage. The higher the stage, the more powers.
Your stage is determined by how hungry you are. Stage 1 means you've just eaten, Stage 4 means you're starving.
I'm not entirely sure what the visual differences are. I know you get paler and your eyes are different depending on your stage. I think your skin also wrinkles a little, but not entirely sure on that.