Russia was already deeply divided between a series of more or less equal-resourced princedoms run by shifting alliances of princes within the Rurykid dynasty (and a few non-dynasts) well before the Mongols arrived in 1220 for one attack on the steppelands of the lower Don and the main invasion in 1237-8. Kiev had lost its predominance as the resources and relative stability of the emerging North-Eastern principality of Vladimir - Suzdal (within which Moscow was founded as a minor town c. 1147) strengthened in the early C12th, shifting the balance of power from South to North, and had no really powerful rulers from the mid-C13th onwards except ones whose power base was elsewhere and who took Kiev over as an add-on to their realms for mainly prestige purposes - most notably Vladimir-Suzdal, which inflicted a major sack on Kiev in 1169, and in the early C13th Galich (the modern Lviv area). The threat of steppe tribes' raiding as a major drag on Kiev's wealth, army's ability to concentrate on its Russian neighbours rather than the steppe, and potential for expansion already existed as a big problem in the mid-late C12th under the Cumans/ Polovtsy in the lands between Dnieper and Don, and was only turbo-charged by the Mongols' arrival.
The pattern for a divided Russia unable to coalesce under one of its states (or even be dominated by one state for more than the lifetime of one energetic and capable ruler) was already set by the mid-C12th, though the Mongol attack of 1237-40 (especially the symbolic sack of Kiev as the end of an era of greatness in the South) and the subsequent drain of tribute plus occasional raids and executions of rulers did exacerbate Russian problems and gave a chance of increasing power to those states that were further away from Mongols raid' main pathways (specially in the NW) and thus acquired large numbers of refugees to add to their manpower and/ or who managed to keep the Mongols happy by acting as middlemen and tax-collectors (most notably Ivan Kalita's Moscow, though it could have been Tver which achieved this if it had had luckier or cleverer rulers and been trusted more by the Golden Horde leaders in Sarai). The main differences to the thrust of Russian development as I see it were the sack and eclipse of Vladimir, thus giving a chance to its sub-divisions of Moscow and Tver to emerge as more than minor towns, and the exhausting drain of raids and steppe turbulence on the South in general, finishing off the chances of Chernigov (which from the 1240s on became a likely long-term victim for either Lithuania or Vladimir/ Smolensk/Tver/ Moscow, whichever of these 4 ended up dominating the North ) and making Galich unable to both hold Kiev and its region to the E and hold back Poland (and later the P-Lithuanian federation) to the West.
With no Mongol attack, we could have got a less raid-drained Galich holding back Poland, even achieving dominance of the central Ukraine nomads as part of its army and so Daniel of Galich and his heirs having the strength to beat Poland in battle - and so no long-term Catholic-dominated Poland/ Lithuania state ruling the Orthodox population as far as Kiev. No huge Polish-Lithuanian confederation for Moscow/ Russia to regard as its main foe, and for it to slowly gobble up in the 1648 ff wars into Peter's reign and then the Partitions ? No ability of Moscow to present itself as 'The Only Champion of Orthodoxy' , even perhaps no relocation of the Patriarchate from Kiev to Moscow aiding this (though a Kiev vs Moscow row could easily end up in the C14th with two rival Patriarchs , one in each city calling the other a heretic or usurper)?
That way we have a bipolar culturally and linguistically Russian world from the C13th to the C17th and less of a chance of one huge Russian state centred in the North , at least until the latter has all the resources of the Volga basin and Siberia and can overwhelm its rival. So a different trajectory for the long-term struggle for the Dnieper region, though the balance of resources would still eventually give the Northern part of Russia a probable victory unless a large Kievan/ Galician state had managed to develop as a modernising power by the 1500s with the help of Black Sea trade links to the Mediterranean world (eg a surviving Byzantium not a Moslem power running the Bosphorus and Anatolia).
In Anatolia, the absence of the 1243 Mongol invasion and the battle of Kose Dagh would keep a large and successful Seljuk state as the region's largest power , though as the Nicaean Byzantines had halted the Seljuks ' advance on the far West in 1211 the full Seljuk conquest of Anatolia is unlikely. The Seljuk state would probably continue to be centred at Konya in the Se, not be based in the NW and close to Constantinople like the Ottomans were - so the Byzantine state re-established in C in 1261 could last longer. The absence of a new flood of tribes into Anatolia might even keep the Ottomans out of the region altogether - they only seem to have arrived in the 1280s. Would the emerging 'Turkish Anatolia' have bothered to send big expeditions to the Aegean coast or the Bosphorus if it had no major accession of 'ghazi' tribal warriors on the frontier and was busy fighting the local smaller emirates, not a huge Mongol Ilkhan state, over NW Iraq and Azerbaijan instead? (The Seljuks had a long-term hereditary claim to rule Iraq and Iran from the 1070s; their main state in the later C11th had been based there not at Konya .)