What are the most significant changes in Russian language in 20th-21st centuries other than orthography rules and lexicon?
The heavily artificial, but apparently catching, trend towards indeclinable brand names and titles (driven by marketers trying to preserve what they call "brand integrity"). They're using a grammatical loophole of a sort, the implicit qualifier:
работа в "Макдоналдс" rather than
в "Макдоналдсе" implying
в ресторанах "Макдоналдс", etc. This has led to occasional examples of people not declining anything that goes in quotes, including film and periodical titles. Personally I'm worried/grumpy about the younger generation picking up this lack of declension as intuitively "correct", but it's not such a big deal in the end.
In other "kids these days" developments, the malapropist substitution of
неприкосновенный seems to be unstoppable; it's all over the media and no one cares. The latter word may go obsolete, and the former officially take on a meaning (untouchable as in immune or inviolable) antonymous to its original and strictly correct one (untouchable as in pariah).
A minor but very curious example of English borrowings influencing the Russian phonetic inventory, further than the already semi-accepted /w/, is what seems to be a palatalised glottal stop: at least one video game streamer on Twitch rendered "early drops" (a Hearthstone term) as
[ʔʲо]рли-дропы (there's currently no way to even spell it standardly); this may become bigger in the future.