Did the Russian Emperors keep the title of tsar in addition to their imperial title? Was the usage of "tsar" instead of "emperor" a slight, perceived or real, towards the Russian state in comparison to Western ones?
There are several things going on here with regards to the persistence of the use of tsar in the West. Firstly, it is sometimes common in historical writing for the weight of historical tradition to keep names and titles in circulation that were not really used by the societies in question (e.g. Byzantine Empire, Kievan Rus'). In this case, it is not so much an issue of a slight, but rather an issue of language and tradition within the historical community. More often than not, Russian historiography tends to distinguish between using "tsar" to denote particular Russian institutions under the empire (tsarist autocracy, tsarist secret police, tsarist Russia, and so on) and reserves "Emperor" for the title of a monarch (Emperor Alexander I, etc.) .
Part of the confusion also stems from the fact that the Romanovs could be of two minds about the title of tsar itself. Peter I accepted the title All-Russian Imperator in 1721 and it became the first title used by subsequent Romanovs. This was part of Peter I's efforts at building up his prestige and self-styling as a modern European monarch who had no peers. The tsar title implied he was still Muscovite, but imperator is a more ambitious. Yet the title of tsar still remained in use and became part of the various monarchical appendages that signified areas over which the tsar had suzerainty, such as the Tsar of Kazan, Tsar of Siberia, etc. The British diplomatic corps' manual, Statesman's Handbook on Russia 1896 advice on addressing the monarch reflected the importance the Romanovs placed upon the power of tradition:
The title of the Russian Monarch is Emperor and Imperial Majesty. Originally the Russian Sovereigns bore the title of grand dukes. With the uniting of Russia under the dominion of Moscow, the title of Tsar began to be used, and was, definitely adopted by Ivan IV in 1547. This remained the title of the Russian Sovereigns until 1721. In 1721, by the peace of Nishtadt, the Great Northern War, carried on so successfully by Peter the Great, was concluded. In celebration of this event, the Senate and Synod resolved to beg Peter I to accept the titles of Emperor, Great, and Father of his Country. A supreme Ukas was issued to that effect, on November 11, 1721, which gave rise to a protest on the part of many European states, as it placed the Russian Sovereign on the same level with the Emperor of Germany, the sole Monarch of that rank then existing. First to acknowledge the new title were Prussia, the Netherlands and Sweden, last - Poland in 1764.
The title Tsar also entered into the vernacular as a shorthand for the Russian ruler and subsequent emperors after Peter I sometimes preferred Tsar over Emperor. Tsar came easier as a form of address for a Russian monarchy increasingly reliant upon Russian history and tradition to legitimize itself in the late nineteenth and early twentieth centuries.