They’re near enough to the same as makes no difference. “Dzi” and “dzu” are old-style romanisations (which still appear in some Japanese loanwords found in English like “kudzu” or “adzuki”).

Lemme rephrase my answer, then: “zu” and “ji” are essentially how they’re pronounced. “Dzu” and “dzi” was how they used to be written in Latin characters, but note that how characters are romanised doesn’t change their correct pronunciation (only how they appear to foreginers).

As a case in point, the word “Peking” (as in “Peking Duck”) is pronounced “Beijing”, same as the city. It’s written in an old-style romanisation called Wade-Giles, and to be pronounced “pee-king” as most foreigners do, it’d have to be written “P’i-k’ing”. Yeah, I know that’s counterintuitive – that’s why we don’t use Wade-Giles any more. Same with “dzu” and “dzi”.

Basically. You won’t encounter them terribly often, so don’t fret too much, and noone’s going to jump down your throat if you quite get it perfect.