Big points about Emperor compared to Monarch...
Barbs start to be a real threat. You can't just ignore them and count on your city garrisons or maybe a panic-whip; you need to have a plan. Either you want a way of fogbusting with warriors (if you can do it with 5 or fewer warriors, this is usually best), or you'll need to get a strategic resource and either axes or chariots - one every city or two.
Maintenance is noticeably higher. Be aware of ways into early gold and beakers - it'll help a lot to get early Writing and an early Library; if you get a chance to, settle an offshore city for overseas trade routes. Financial coast and/or GLH are also reasonable options. Consider some early cottages (especially around your capital). Use tile overlap to let them grow non-stop without forcing any one city to always work them. If you're rushing, don't automatically keep all the cities you take - if there's a city in a bad spot, just raze it. Settling compactly except for blocking cities or cities that take gold/gems/silver will help a lot.
While on the topic of rushes... they're harder on Emperor. The somewhat later "rushes" (horse archers or catapults) start looking a lot stronger compared to the early axe/chariot, although on Emperor I'd still say the early rushes are common options.
The AI starts to compete for stuff - you have to make a legitimate effort to get wonders, first-to-tech bonuses, and such. On the flip side, this means you can be more confident relying on stuff like failgold, trade-bait techs, and support from vassals (on Monarch-, you just can't count on the AI to do it's part on any of those efforts).
Early happiness starts to be a serious concern. You should have a plan in mind for getting at least 2-3 extra happiness in your cities fairly early in the game. If you have gold and gems, great; more happiness is nice, but you can work with what you have if needed. Or you may end up getting a religion and temples, building Pyramids, getting Monarchy early... easiest way to suddenly discover that you've been falling behind for the past hundred turns is to forget to find early ways of raising happy cap.
Score isn't important, ignore it. Here are the fundamentals in my judgment - do you have enough good land? On IMM/Deity you often have to just man up and live with the fact that some AI empires will be bigger than you for a long time, but on Emperor early aggressive expansion through REXing or war can usually get you at least as much good land as the AIs are taking. Are you keeping up in tech at least enough to trade? As long as you're in shouting distance of the AI, you can stay competitive despite a weak research rate for quite a long time (during which you can develop your land, take new land, get key wonders, manipulate AIs into costly and research-slowing wars, and otherwise adjust things to let you move to the lead in research). Is your diplomatic situation tolerable? If you've managed to become worst enemy of several empires, all of them warmongers that are larger than you and ahead in tech... well, you messed up somewhere and will probably lose. If there's no way anybody could plan on attacking you, or you are strong enough to easily destroy anyone who might attack you, the game is in pretty good shape even if you are behind in score, land, and research rate.
If the answer to all those three questions was yes, I wouldn't be concerned even if every other AI somehow had twice my score. If the answer was "no, but there's something I will be doing soon to make that answer a yes," I wouldn't be concerned. It's only when you're so hopelessly behind in land and tech that you cannot trade and cannot fight a war, or when you're expecting a doom-backstab any second, that the game starts looking grim.