The difference isn't large like from Emperor to Immortal.
It is easily surmountable. IF you can win King easily, you will probably get the hang of emperor in like 2 to 3 games.
Tech side: The AI starts with Pottery if i am not wrong. Thus they are more than likely to secure a pantheon first and will most likely grab a religion before you if you are not Maya or Ethiopia or you did not manage to find a religious CS.
GL on average is gone on turn 47 so if you have been incorporating GL into your games, expect not to get it half of the time and modify your gameplan to not rely on GL as a clutch.
HG and Petra can be hard to get. CI is gone like 99% of the time unless you beeline CS.
Though it is not uncommon to be tech leader by late Medieval. You can try Babylon or Korea to start off if you are worried.
On War: AI has slightly more gold and has slightly more units but nothing to hard to deal with. Shaka is still a monster.
King to Emperor is one of the smallest steps in the game ... unless you are building too many wonders. Like if you try to build Great Library on a low hammer start.
The AI starts with Pottery if i am not wrong.
I wish I made the jump from king->emperor earlier because I found emperor to be easier than king. On king, you can get away with building more world wonders, which isn't a very good strategy as it's usually better to take them by force (except maybe for the few that have founder bonuses.) On emperor, I found diplomacy to be much easier. Since the AI starts with increased bonuses, It takes a little while longer for you to catch up (late medieval instead of late classical/early medieval) during that window, most of them offer DoF's (except for the REAL haters, mostly just Shaka and Lex) which you have the option of trying to maintain for the majority of the game.