South Ossetia and Abkhazia, as well as Transnistria, East of Ukraine and Crime, have diminished Russia's (so as Russian people) image in the eyes of not just the Caucasus, but also Belarus and Central Asia.
As a Central Asian myself, I do know that people in my home region do not trust the Russians. For instance, in Kazakhstan they have started a migration program to attract more Kazakhs to move to the North (heavily Russian population area) in order to improve the issue with the aging population there. Well, we all know that they are doing it, just because they don't want to go through the Crimean scenario, if things go rough.
Belarus president had stated numerously that he does not believe in 'historically my territory' claims. Although he had acknowledged that Crimea is now part of Russia, he did in fact mentioned that for him, it is important that other states respect each others borders.
Azerbaijan had drifted towards Turkey since 1990's. After they have lost in the Karabakh War, they realized what kind of threat does Russia posses against those, who do not comply with their conditions. For that reason, not only did they just condemn Russia's actions at the UN council (back in 2015), but they also have developed their relationship with NATO.
Armenia is the only pro-Russian state in the Transcaucasian region, but even now, with the new government in place they might experience certain political modifications in their country.
As for the question of importance of South Ossetia and Abkhazia, to be honest, in my opinion they are not important at all. Not even strategically. Russia only helped them, just to show their military and political might over the countries of the post-Soviet Union. They wanted to show them that the West will not hassle, but Russia will do whatever it takes to implement its authority over the others.