RS3: Micro transactions, better graphics, totally different combat system with action bars, screen size, everything is easier and afkable, and double exp etc events.
OSRS: Old school graphics, much more grind oriented, smaller screen (hopefully resizable sometime this year), smaller community, generally slower exp, auto combat system basically, no double exp events, and a community that wars with each other on what updates the game needs. Oh, and OSRS is for the most part community driven content via a polling system.
I play RS3 and don't play 2007scape but I'll try to give as best of an answer as I can.
2007scape is basically it's own game now. Sure, same company and similar play styles but its updates are all different than the main game. Its classic combat with no abilities. There have been various nice updates such as the grand exchange, new bosses, new minigames and such.
Their PvP community is very strong, and all levels generally will be able to pick a fight. PvM is being improved periodically with the introduction of new bosses.
Some say that OSRS is about nostalgia, others say that its about rebuilding RuneScape from its subjectively negative updates. Its underlying problems are criticisms within AFK training in a minigame called nightmare zone, drop tables, and AFK magic training via splashing just to name a few. RS3 has a more complex combat system. Same grid based movement, but you can put things on action bars. There is a Legacy mode that reverts and scales everything back to the old combat style, but is far less effective than the ability based system. Skilling is overall more fluid in the sense that there are more quality of life updates. Skilling is generally less click intensive.
RS3 has more content that appeals to end game PVMers. Its underlying problems are micro transactions infesting the game, content output (as of late), and pay to win aspects. There's dead content everywhere. The PvP is pretty much dead from what it was years ago. New players are still coming, but with how deep the game is and how a good amount of the people have already hit "endgame", new players can easily be shyed away. There's my input. I may have forgot alooot of things, but hopefully this gives you a sense o