A detailed discussion of Russia as it exists today is probably outside of the scope of this sub. However, it's not difficult to highlight differences between the USSR in its prime and the Russian federation post-disintegration.
The USSR was a larger and more populous entity than the Russian Federation. The USSR's population on the eve of the dissolution was 290 million. The population of the Russian Federation in 1991 was approximately 150 million, a figure that it is still at in 2016. This is close to a 50% loss of population when the Russian SFSR struck out on its own. The USSR was not just Russia, it was Russia, Ukraine, Khazakstan, Moldova, the central Asian republics and the Baltic States. After the dissolution of the USSR, these countries experienced the effects of a demographic decline that was in the making late in the USSR and has not really recovered until recently in Russia.
The USSR possessed a variety of client or puppet states in Eastern and Central Europe. Any conflict with the USSR would drag the other Warsaw Pact countries into the fight as well. These countries maintained large military forces in their own right. Before the USSR had dissolved, these countries had already started breaking away and seeking ties with the west.
The USSR had endemic economic problems, and these started becoming major problems by the end of its lifetime. Russia inherited the economic situation of the Russian SFSR, and was subsequently exposed to additional economic shocks as it transitioned from a planned economy to a free market economy.
The USSR in the 1950's and 1960's was on par with or ahead of the United States in some areas of technology, particularly military technology. By the 1990's, the United States had pulled well ahead in essentially every area. The economic situation in the early Russian Federation was extremely bad as mentioned above, and this meant that large scale military R&D was out of the question. Even maintaining forces was not possible, and the size of the military as a whole shrank, even beyond what was lost when the other SSRs broke away and formed their own military forces. Naval forces were in particular disrepair, as they had not been the top priority even in the USSR's heyday. This is relevant when we speak of power projection especially.
Russia in 2016 is still a very powerful country. But it lacks the economic power, the technological sophistecation and global power projection capabilities to challenge the United States in a meaningful way outside of its own back yard. Its large nuclear Arsenal is the most visible vestige of its superpower past, but in other respects it is a shadow of its former self, economically, demographically, militarily and geographically.
I think you'll find /r/geopolitics to be a better community for this question.