The difference between these four is the topic of the article on yotsugana (四つ仮名). In south Shikoku, they appear to be all different. However, in standard Tokyo Japanese, じ = ぢ and ず = づ.
The actual diachronic change of [dz] to [z] is variable in standard Japanese. According to "The Sounds of Japanese", the affricate [dz] occurs at the beginning of a word or immediately following a syllable-final consonant, and the fricative [z] occurs in the middle immediately following a vowel. This isn't affected by which kana is used to transcribe it. The kana じ and ず are more common anyway since the gendai kanazukai (現代仮名遣い) post-WW2.
As for Rōmaji input, I've always used t + u / s + i with dakuten.