How different is the life of Moscow from the rest of Russia? Is it true that the rest of Russia lives much worse than Moscow? What can usually be seen in Moscow in terms of tourism?
It is different, people go live and work in Moscow primarily for $$$. Russia is different, and Far East, Siberia, Urals, Altai, Carelia, South Regions, etc. all have their peculiarities and cultural differences.
In terms of $$$ it is true that outside of "economically active" cities life in Russia is "worse" (and cheaper). In terms of "union with Nature", it is better, farther you are away from these cities.
In Moscow you can primarily see the Kremlin, some Cathedrals, and Metro - underground subway system that has some old stations that could be of interest. Moscow State University and surrounding area, probably. Some "modern architecture" in Moscow-City business center which is aplenty in the West countries. It is all boring though. You can find lots of guides and photos online.
St. Petersburg is more rich in terms of architecture and is not as diverse as Moscow.
Advice: better not to travel outside of Moscow and St. Petersburg without having local currency. Exchange in airport upon arrival. ATMs in rural areas could be absent, or reject your card, or do not have enough cash, or have substantial withdrawal fee. Also English in rural areas wouldn't be sufficient to converse with locals. Not be too trusting to strangers offering "services" at airports, train hubs, and other such public places.
If you are going to visit soon, you could contact me privately via Skype or similar for some advises. There are others living in Russia on the Forum who could guide and even host you if you decide to come for a visit.