Okay, I'll preface this by saying that it's basically impossible to know for sure. But, one big observation to make is that both Zyuganov and Zhirinovsky as well as Yelstin, although to a lesser extent were running very nationalistic campaigns, and while the exact policies of Russia would have been different between the Communists and the LDPR, Russia's position in the global system probably wouldn't have waivered much from the path it took in reality.
Now, looking at their respective voting blocs - electoral demographics and geography have changed little since 1996. Despite United Russia emerging as the dominant political force across the country, communists continue to do well in the west, the grain belt, and in industrial cities. Meanwhile, the nationalists tend to do better in Siberia and other remote areas - basically the areas Yelstin won in 1996 and where Putin and Zhironovsky over-performed in 2012. I made a map of how the parties did in in 2012 relative to their national total a while back for a different thread:
United RussiaCommunistsLDPRProkhorov
A Just RussiaSo that gives you some sense of what the map might have looked like. Now, in order to determine who would have actually won, I think it's helpful to look at how some of the other post-Soviet states "voted" in the 1990s - in a lot of cases, there really wasn't much of a vote, but it's still useful to see what kind of government emerged:
Ukraine - Economically liberal kleptocracy, generally pro-Russian
Belarus - Dictatorship under a largely pro-Russian, neo-Soviet, nationalist, agrarian banner
Moldova - Agrarianism followed by communism
(Transnistria) - Neo-Soviet pro-Russian nationalists, similar to Belarus
Georgia - Basically mob bosses running various factions of the country
Armenia - Economically liberal, nationalist kleptocracy
Azerbaijan+Central Asia - Big tent nationalists
So, except for Moldova and arguably Transnistria and Belarus, the rulers in the 90s were generally closer in profile to Zhirinovsky than Zyuganov. At the same time, few of them were anywhere near as extreme in their nationalism as Zhirinovsky was, but perhaps this has to do more with the realities of governing than a general rejection of ultra-nationalism.
All in all, I think Zhirinovsky would have had the upper-hand, but it would have been close. In 1995 the communists had a fantastic showing in the Duma elections compared to the LDPR, and, of course, Zhirinovsky ended up finishing well behind Zyuganov in the first round of voting. If he had made the second round, however, I think a lot of anti-Zyuganov forces and Yeltsin voters would have coalesced around the LDPR. But again, I doubt the direction of the country would have radically changed either way.
And, of course, this whole discussion is meaningless, as we all know Zyuganov would have won on account of his baldness.
