By "Roman plate" I assume you mean lorica segmentata, which is what you tend to see a lot of in popular culture. We're not exactly sure which Roman troops used it, or even how the Romans themselves called it, but it was used from 1st - 3rd century AD. Lorica hamata (mail armour) is what most Roman troops wore throughout Roman history. As for why, it's because it is more costly to build and maintain than mail armour. After the crisis of the 3rd century there wasn't enough economic power to sustain it.

There was less continuity between Greco-Roman arming traditions and the Medieval period than there was between Germanic and later Carolingian arms and Medieval Christendom.

Late Roman armies mostly used chainmail, as well as their Germanic successors.

The early middle ages was dominated by chain mail, and plate armor didn't become common until after the high middle ages- around the 14th century. Complete plate- covered joints and the like- wasn't even a thing until the late 1400's.

Timeline of the development of Western armor from 650-17th century.

Plate armor developed because of advances in metal shaping technology and novel weapons developments- particularly more devastating missile fire (longbows could pierce chainmail in the 1300s, steel-string crossbows in the 1400's+, and finally gunpowder). Hand-to-hand combat also changed, with short swords meant for thrusting (an extreme example) eventually becoming the norm by the late medieval period (two handed swords remained but were developed for specialized uses in pike combat). Rapiers and other piercing swords became more common, though some larger cutting swords made a comeback in the 17th century.

Follow up question: How did plate armor go from basic infantry armor in the form of Greek cuirasses and grieves to the most expensive and protective armor like that used by Henry VIII?

Edit: Added detail.

All "plates" are not made equal. Just because the Romans use a type of solid metal armour does not make it immediately similar to late medieval plate, as the technology involved had advanced quite a bit. The invention of the blast furnace in Europe allowed armourers to produce high quality plates cheaper and in a fraction of the time it took to produce a maille shirt. It's economics basically, when the Roman slave factories could no longer produce lorica segmemtata at a profit they fell back on the time consuming lorica hamata. A slave's time is a lot cheaper than a armourers.

As others have noted it was mostly a change to large scale use of chain mail (which the romans also used). This has several benefits over plate: It can be easily "tailored" to fit different wearers without losing integrity, even by people who don't know how to work a forge, it can make use of relatively low quality iron an still be effective (to make large pieces of strong plate you need relatively pure steel, not just iron), it is excellent at protecting against slashing weapons and missiles, perhaps even better than plate at protecting against stabbing weapons since it doesn't have joints, and it is much lighter.

The main reason the high middle ages turned to solid, largely steel plate, was because of improvements in missile technology most notably the bodkin arrow and the crossbow. These were designed to pierce mail, so if you were going to war and could afford it you now wanted solid steel plate to stop some peasant from killing you with an arrow you as you charge.

Additionally, technology did develop over the course of the period in question say 0-1600AD, such as the blast furnace, bloomery furnaces for iron tools, which made steel more accessible and mines more productive.