So there's a lot to compare here in this fascinating contest of Who's the REAL Rome Now? I'll be editing parts from a previous answer here.

​

Let's start by looking at the origin of both states and how they viewed themselves and each other, the Empire of Tebizond and what you accurately coined the Late Roman Empire, the state that existed after the Fourth Crusade sacked Constantinople and splintered the empire into three states, the Anatolian heartland of Nicaea, the eastern coast of Epirus, and the Pontic state of Trebizond. Actually, though, Trebizond predates its two brother rump states. Part of the Crusader justification for camping outside Constantinople was to place their claimant, Alexios IV Angelos, on the Imperial throne. Besides the violence in and around Constantinople, this succession battle spread to the outer provinces as well. In April 1204, mere days before the city would be sacked by the crusaders, David and Alexios Komnenos, grandsons of former Emperor Andronikos Komnenos, captured the wealthy Black Sea trading city of Trebizond and proclaimed themselves Emperors. Brought up at the Georgian court after the Komnenos family fled the city following Andronikos’s death two decades previously, the brothers held a strong claim to the Byzantine throne through their families’ legacy. So at least at first, the Trapezunite rulers styled themselves as the true claimants to the Roman mantle. However, this claim was not to last. After the Nicaeans reconquered Constantinople, recognizing that their power had risen beyond their own and in order to ease tensions between the states, the ruler of Trebizond John II Komnenos renounced his claim on Constantinople. He and future rulers of Trebizond instead styled themselves Emperors of the East, Iberia, and Perateia, still Imperial rulers, but not *the* Imperial ruler. My colleague u/nikvelimirovic also pointed out in the previous thread that "while they renounced their claim on Constantinople, and thus implicitly acknowledged the Palaiologoi as the continuation of the Roman Empire, they continued to style themselves as emperors, and using the title Μεγαλοκομνηνοί (Great Komnenoi)."

​

In contrast, the Empire of Nicaea, from the outset, through the conquest of Constantinople, and up to the final fall to the Turks in 1453, claimed to be the true Roman successor. Retaking Constantinople from the Latins, defeating the Epirots, and negotiating with the Komenoi of Trebizond, all served to bolster this claim, though the West refused to acknowledge its rulers as anything other than "King of the Greeks." So with regard to political perspective, both states maintained very different claims and identities, and many of their actions stemmed from this. While both continued to carry out the famed Byzantine diplomacy, paying or playing their more powerful neighbors against each other whenever possible, Trebizond tended to be more localized whereas Nicaea->Late Byzantium exerted a slightly greater reach, at least earlier in its history.