For Finland it was early on very much the same as what it had been during the period Finland was the eastern part of Sweden. The reason for this was the extensive autonomy granted to Finland. While early on the actual governing structures were rarely assembled the autonomy helped to preserve the Swedish era system. Finland also somewhat expanded at that time as some of the lands of Swedish Finland that Russia had gained in the peace treaties of 1721 and 1744 were turned over to the Grand Duchy of Finland - which lead to several issues since for example the Russian nobles who had moved in had brought serf with them and those were illegal in Finland.

Finland continued to have extensive autonomy with local government, own currency, separate custom & tariff system, its own army, postal office, Swedish (later also Finnish) as official language and so on. So while that lasted there was very little change from the situation that had existed in 1809. Main problem in a sense was that the Russian system also prevented any reforms being made for the first 50 years of the autonomy - unlike in Sweden. However this situation also created problems since Finns were considerably better off than what Russians were and Russians (or other subjects of the Russian Empire) were not even allowed to move in to Finland without a valid reason (like military service).

The final two decades of the Russian rule in Finland were however very different from the above. In that period Russia - inspired by Russian nationalistic ideals - Russia started trying to accomplish the Russification of Finland. That in Finland the two periods (1899–1905 & 1908–1917) are known as the 'times of oppression' (sortokaudet) is probably telling enough. This was in essence an attempt to destroy the Finnish autonomy and turn it into a part of Russia. It didn't achieve that - mostly it only managed to drive a massive wedge between the Finns and the Russians.

So i could not say that life in Finland would have been better under the Russian rule - at best it was about the same as what it had been during the Swedish rule because the Swedish rule was in essence retained - just that instead of a Swedish king there now was a Russian tsar as the head of state. Russia did sponsor the growth of a separate Finnish identity from that of Sweden to make sure that Finland would not at some point 'defect' from Russia back into Swedish control and therefore paved the way for the Finnish nationalism and also somewhat ironically for the Finnish independence. This also meant allowing the importance of the Finnish language and the Finnish language based culture to grow and expand (like becoming second language of government and so on). However while in cultural sense and the sense of developing national identity such events were extremely important they didn't really affect the life of the ordinary people at the time.