I dont see why a russian empire couldn't be as developed as the rest of northern Europe or at least Italy and Spain by the modern day. Being integrated into the world economy for 100 years instead of the horrifying experiment of OTL, would surely do wonders for developement. It would remain a very socially conservative country, with its peasant roots still strong, for better or for worse.
If it were to survive to this day, i would say it would have a slightly lower GDP than US, but a slightly higher population, maybe 375 - 400 million. Ukraine and Belarus wouldnt exist. Ukrainian nationalism basically only existed as a force to be reckoned with because of Austrian and German funding pre wwI funding, and Ukrainians were happily part of the empire, until WW1. Ukraine would have a massive population, more or equal to Germany. Kazakhstan would likely be majority ethnic Russian, with the Kazakh language under threat of disuse. If the Empire still exists, and had lost no major European wars, then poland would be tiny, and germany would still have more/less 1920 borders. I wouldn't be surprised if Latvia and Estonia would be majority russian, probably not Lithuania though. I would say the Baltics would remain in the empire. Transcaucasia would brake away, but north caucasus would be majority Russian. Unfriendly relations with giorgia and Azerbaijan, but allied with mega armenia which would have absorbed massive chunks of Anatolia during Turkish war after ww1. The straits would remain russian until maybe 1970s when they would be transferred to a Greece in possession of the whole Aegean coastline. Rump Turkey with no friends, which most likely embraced some sort of fascism in 20th century.
European countries would be 10x friendlier with russia than OTL, having stronger bonds and cultural links than with the US. China would be the largest economy, having surpassed US and Russia in 1990's. Russia wouldn't be a liberal domocracy. There is a reason the vast majority of russians accepted autocracy until 1917 and again until end of soviet union, and then so readily accepted Putin. Patrimonialism is highly embedded into the russian psyche, its roots in the Muscovite traditions which lasted centuries. I would guess it would be a very illiberal democracy, much like Hungary today, with a strongman priminister running the country, with help and legitimacy from a supportive Tsar. Orthodox Church would be hugely influential, Catholic faith would be bigger than OTL in Russia.
Siberia ITTL would be way more heavily populated than OTL, with a 'wheatbelt' streatching from urals to the transbaikal region. Russia would be wholly self sufficient in almost every manner, including oil, gas and foodstuffs. Major cities would dot siberia, Maybe Novonikolaevsk or Irktusk would have 3 million plus population. Moscow would be the biggest city and industrial centre, while St. Petersburg would be the political and financial centre.
Architecture and art would be much better than OTL, without repressive government and WW2 destruction. Stalinist and brutalist architecture, thing vast grey apartment blocks, wouldn't exist.
Science, rather than just excelling in highly niche miliary and space areas like physics, would be much more diversified like the US and Europe. They may or may not have a space program equivalent to OTL, although if they did, it would happen much later than OTL due to no cold war.