Russia without Vodka? This is ASB.
Seriously, Russian culture would develop very differently - instead the main influences being being Greek and Church Slavic, they would be Arabic, Persian, and Turkic. As Russia didn't have an own literary language at that time, this place would be taken by Arabic and / or Persian, leading to a much wider separation between the literate elites (clergy) and the non-literate (Church Slavic and Old Russian were very similar, which is not true with regards to main Islamic literary languages at the time). It would also lead to increased animosity between Russia and its neighbours to the West and South-West, Poland, the Bulgarians, and Byzantium.
If we assume that other major developments stay the same, here are some of the possibilities:
1) Russia may become a target of the Crusades. Of course, IOTL Russia also had its conflicts with the Teutonic order, but ITTL this would be worse and perhaps more Christian nations would be involved. It also may mean that Western crusaders would concentrate less on the Baltic shore.
2) If we assume that the Mongols come conquering as IOTL, they probably would crush Russia as well. But without the antagonism between the Islamicised Tatars and the Orthodox Christian Russians, assimilation to the Turko-Mongolian political system may be even stronger, with Mongolian dynasties taking root in Russia, Russian rulers using Mongolian titles like Khan, etc. Again there would be stronger antagonism with the Western neighbours, and the main goal of Russian rulers would be jihad against the pagan Lithuanians and the Christian Poles etc., not the liberation from the Tatar yoke. Russian independence would be gained by Russian Khans asserting their independence from the Steppe-based Tatars, but as the main enemy would be the Western neighbours, there would be less of an ideological background to this.
3) An Islamic Novgorod would be less open to the West and would play less of a special role; it would be just one of the Russian territories not directly ruled by the Volga Khanate.
4) The Lithuanian expansion of the 13th/14th century would probably run into resistance earlier. With the stronger polarisation between Russia and its Western neighbours, the Lithuanian elite would have to choose early on between Islam and Catholicism.
5) If we get a united Islamic Russia, it may become an ally of the Ottomans, at least until their borders meet and they become rivals.
6) Probably Russia (or Russian Khanates) would still expand East to Siberia - empty spaces are empty spaces.
7) There would be less of a difference between Cossacks and the Muslim steppe tribes. That may either hamper or speed up Russian conquest of the Steppe.
8) As an Islamic state, Russia may have more problems than IOTL with opening up to and implementing Western ideas and reforms. Its modern history (17th century onwards) would probably be quite different.
9) Without the bond of the Orthodox church and the "3rd Rome" ideology, Russia will be less interested in the Balkans.