Well, after 1204 the Byzantine Empire was partitioned. The Crusaders formed the Latin Empire, centred on Constantinople with three Greek rump states each claiming to be the legitimate restorer of Byzantium. The first, the Despotate of Epirus, ruled by a distant branch of the Doukas family, made a good stab at taking Constantinople back but then fell apart. The second, the Empire of Nicaea, ultimately ruled by the Palailogoi, managed to recapture Constantinople and refound the empire. The last successor state was of course Trebizond, ruled by the Komnenoi the imperial dynasty prior to 1204. The reason why it called itself an empire therefore is that it claimed to be the legitimate successor of Byzantium, and was therefore an empire. However, after the Palaiologoi re-established Byzantium they gave up the full imperial title ""Emperor and Autocrat of the Romans" for the slightly lesser one "Emperor and Autocrat of the entire East, of the Iberians and the Perateia".

Trebizond essentially existed as a regional power until the Ottomans came along, often allied to Georgia through alliances. Dynastic troubles and invasion led to the gradual shrinkage of the Empires territory. Despite the best efforts of the Emperors at making alliances in the west and with neighbouring Muslim powers Trebizond was conquered by the Turks with relatively little effort in 1461, a decade or two after the fall of Constantinople.

The last Emperor, David, settled in Adrianople in 1461 living off the profits of his estates. Mehmed had him, his three sons and a nephew imprisoned on a pretext two years later, brought to Constantinople and beheaded.

The main source of their wealth was that Trebizond sat at one of the ends of the Silk Road, if you wanted to put the east's riches on a boat you did so in Trebizond or Crimea and the emperors of Trebizond profited greatly from taxing trade routes.

I feel it necessary to add one could easily argue that the fall of Trebizond was not the 'final nail in the coffin' for 'Rome' [realistically, the Byzantines], rather Gothia, or the Prinicipality of Theodoro, which outlasted Trebizond by 14 years (until conquest by the Ottomans in 1475). The state was ruled by Alexander of Theodoro, with a population varyingly comprised of Crimean Goths, Greeks, Alans and a few other ethnic groups. It served as a splinter state of the Byzantine Empire in a similar manner to that of Trebizond, albeit of lesser status.