What is it about the large boards that make it difficult to win on Regent? That might answer some of your questions.
On Monarch I have found that you rarely have a lead (or much of a lead) in techs anymore. The AI researches better and trades better (cheaper at least) with each other. So you are often fighting for tech parity more often. IMO, it is trading skills that are mostly impacted. Self-research is not impossible, but it much harder. You actually need to accept that some AI civs are on your world, at least for a while, and trade with them.
Also, if you haven't worked on your REX game, you will need to step that up. The faster production of the AI means that they will be filling up the land much quicker - so you need to be on top of it. Settler and worker factories - you need to learn how to use them. It is a huge advantage over the AI during REX.
You are less likely to win a 100K cultural victory (on accident) because the AI will build just about every cultural building there is. Flips are also more likely to occur because of AI culture, something you rarely need to worry about at Regent or lower.
You will have less units than the AI (but you will make more workers, right?). That means you need to fight smarter. You can't just throw your units away because you outnumber the AI 3:1. Now they will outnumber you.