Summary
Primary difference is that Kripke style uses context stacks while Fitch style uses locks. But this difference is syntactic, and primarily philosophical, not technical. One can probably find an easy correspondence but they have different sources of inspirations.
Kripke style, as indicated by its name, is inspired from Kripke semantics. On the other hand, Fitch style is popularized by Clouston, who followed Borghuis. In Borghuis' PhD thesis, he derived modal pure type systems (PTS) from Fitch's subcoordinate proof diagrams. Borghuis never used Fitch style to refer to a type system. He only used it to refer to that particular kind of subcordinate proof diagrams. Technically, Fitch in Fitch style really denotes the particular proof diagrams or form of proof presentation which Fitch invented and Borghuis considered. However, Fitch-style systems do not really have this form of subordination in their formulations, nor do Fitch-style systems use context stacks as Borghuis originally did (he called them generalized contexts). This makes Fitch style more or less a historically inaccurate name, at least less accurate than Borghuis style.
TODO: More Comparison