I came. I saw. I conquered.
I came; I saw; I conquered.
I came, I saw, I conquered.
All grammatically viable, but the voices are different. Punctuation, as I have said before, is all about signalling in what voice something should be read. So here, for what it's worth, is my ha'penny's worth.
The way I reckon it the full stops make it sound as if you're marking the sentences out with your hands: perhaps banging three times on a conference table with your big hairy hands, perhaps thrusting your index finger towards your interlocutor's face on each verb. Maybe spitting slightly. It's how I would write out a Hitler speech. I think all the words would be delivered at the same pitch.
The semicolons, on the other hand, make it sound like Churchill. It's being said slowly, but with a single cadence running through it. Came would be highest pitch and conquered lowest. I think there could even be a bigger pause on the ;s than on the .s.
The commas make it sound like Bertie Wooster: a sort of devil-may-care off-the-cuffishness. 'What did I do today? Golly, Brutus, I'm not sure. Oh, that's it. I came, I saw, I conquered. That sort of thing, I suppose. Care for a drink?'
That's how I read them anyway. Many people are worried about the near extinction of the semicolon. They sit at home and weep over its absence like a doting mother whose octuplet sons have all run away to sea. The reason for ;'s apparent scarcity is, I think, not that people don't know how to use it, but that most don't want to sound like Churchill. They have no ambition, you see.
Most style guides would have you write like Hitler.
It's also worthwhile remembering that the first writer to really use the semicolon was Ben Jonson. Shakespeare and Chaucer seem to have to got by without. The ; was invented by Aldus Manutius, who also invented italics.