No polymorphism for the vampire is detailed in 5E
The entry for vampire in the Monster Manual p 295 doesn't really differentiate on what the creature's previous race was. This system assumes that, short of DM fiat, humanoids can become vampires however their race immediately becomes undead and the stats are the same as that listed in the MM. Some DMs such as myself view Vampirism as a mode of possession, as portrayed in Order of the Stick and "Buffy the Vampire Slayer", which means it really doesn't make any difference what your race was.
AD&D 2nd Edition
Now in certain settings there was some polymorphism introduced. Most notably in the old Ravenloft setting. In the Monstrous Compendium Ravenloft Appendix (1991) from 2E.
In this source was detailed variations on the abilities of the standard races if they were afflicted with vampirism.
Elves: Couldn't go out at night and could wander around in the day time. Teleported through plants. What amounts to a gaze to paralyze. Their touch withered and killed plants. Unaffected by Holy Water but could be burned by contact with diciduous plant sap.
Dwarves: Could essentially Earth Glide which made them particularly scary and difficult to find their lair. Instead of a stake you had to use a natural stalagmite or stalactite. Could not cross a lines of powdered metal. Could change to a burrowing animal instead of bat and wolf.
Gnomes: Could not turn away from a jewel. Could only be hit with metal. Could not shapechange. Had a spectral form rather than gaseous. Gaze attack to inflict Tasha's Laughter effect.
Halfling: Repelled by pipe smoke. Rain was deadly. Shapechanged to woodland creatures and could control them (beavers, squirrels etc.). Create food and drink (lures prey).
Just as examples of the differences they highlighted. This gave them flavor.
Another great source are the Van Richten's guides from the 2nd Edition Ravenloft setting written by the fictious NPC Dr. Rudolph Van Richten, these have great flavor details on how to use vampires as well as salient abilities, which are powers that vampires get as they age and gain more power. 2E is pretty compatible with 5E as far as the rules are concerned and the flavor text is universal. I still use these old books when doing more horror driven stories where the primary antagonist is a Vampire, Werewolf or even a Golem in some cases.
Dungeons and Dragons 3.X
Templates from 3.XE also allowed for differentiation. I do miss the templates from that system, gave a great many options even if some of them were down right silly. The template for vampire didn't really change with the race it was applied to however, it simply added the abilities of vampire to the existing creature, in my 2E example the abilities of the vampire changed based on what was being possessed or afflicted (take your pick on how you define vampirism).
Real World differences
There are differences in how a vampire is viewed in the real world as well. If we throw out sparkly, emo, stalker vampires from popular culture we have a number to choose from:
There are two versions in Asia to my knowledge. One is invisible all the time or to just some people and the one from the Philippines is called the Aswang in Tagolog and Wakwak in Bisayan. The latter amounts to a shapechanger, human by day and at night flies on wings with a long tongue.
Then of course there is the original popularization from Bram Stoker.