Just to add some more information..there are some antigens which may afford some level of cross protection. But by no means would this be complete. Clinically you can find patients with both viruses reactivating at the same time.
The majority of patients with HSV-2 meningitis lack HSV-1 antibodies in serum, so there is a protection against HSV-2 meningitis if you are HSV-1 antibody positive previously ( Long-term Valacyclovir Suppressive Treatment After Herpes Simplex Virus Type 2 Meningitis: A Double-Blind, Randomized Controlled Trial
E. Aurelius,1 E. Franzen-Ro ̈hl,1 M. Glimåker,1 O. Akre,1 L. Grillner,2 C. Jorup-Ro ̈nstro ̈m,3 M. Studahl,4 and the HSV-2 Meningitis Study Group, Clinical Infectious Diseases, 2012 )
Also in genital herpes preexisting iimmune response to HSV-1 is a marker of protection against HSV-2 genital disease (Langenberg, N Engl J Med 1999)
Purely from a clinical perspective on genital HSV infectionsalone, HSV 1 antibodies from previous oro-labial infection does indeed have a protective effect against subsequent HSV 2 infection in that, the HSV 1 antibodies appear to mask the syptoms of a first episode of genital HSV 2 infection to varying degrees, from a very mild first episode to one which is not recognised at all. This may also explain why the majority unrecognised HSV 2 infected Carriers are asymptomatic and vouch to be asymptomatic when their previously HSV uninfected partners develop symptomatic primay genital HSV infection.