Are Symbiotic and Vampire Stars the same type of star or different? Are Vampire Stars and Symbiotic Stars the same type of star or is there a difference?  

Vampire Stars siphon off hydrogen from its neighbour.  According to https://www.cfa.harvard.edu/~kenyon/ss/, Symbiotic Stars seem like the same description.  

"In all systems, the hot component - a main sequence star, a white dwarf, or a neutron star - accretes material lost by the red giant." (Harvard)

SS Leporis has been described as Symbiotic Star and in some places as Vampire Stars.
As the title of the research paper linked to in the ESO press release is "An incisive look at the symbiotic star SS Leporis", it is safe to conclude that they are the same type of star. My suspicion is that, since symbiotic stars transfer and accrete mass via either Roche-lobe overflow or stellar wind overflow, the term "vampire star" was coined as part of the press release as a "cute" way to imply that one star is "feeding" on another.