The "H" in the 308H technically indicates a "controlled" carbon content, with the AWS Specifications stating 0.04 to 0.08% and the product analysis yielding 0.05% carbon. The "normal" 308 specification calls for 0.08% maximum carbon content, with no minimum given. The product analysis for 308 is normally 0.05%, but could technically be 0.03% or even less.
This filler provides more robust mechanical properties in higher temperature applications, because of the increase carbon content.