To put it simply, one of the most noticeable differences lies in the "demand" vs "feeling." Example, きっと来て下さい vs. ぜひ、来てください The sentence using きっと is stating for you/he/she/them to come with a bit more demand "you definitely have to come" - obviously the choice is still up to the recipient but the expression of the demand is there, whereas the sentence with ぜひ is expressing a feeling, something like, "by all means, please come." I hope that makes sense and helps. ps. Of course there are more cases but this is a good place to start.

ぜひ shows a speaker's willingness to make it happen, and is only used for a person. On the other hand, きっと indicates speaker's assurance. Also, when きっと is used in a no-negative context, it, in many cases, implies the speaker's hope to come about.