I would say the 3 biggest difficulty jumps are:
1. Immortal to deity (by far)
2. Prince to monarch
3. Emperor to immortal
1. is a HUGE leap, perhaps greater than settler to immortal, for reasons mostly stated before. The free settler means AIs automatically begin with
twice as much production and food potential, as well as land, which in turn speeds up their proceeding expansions by an exponential degree. On most deity games, AIs found their 3rd city faster than they found their 2nd on immortal, both of which are faster than
you found your 2nd on any difficulty noble and above. Besides that, religion also plays into their favor because founding a religion means a free border pop (2, eventually) and high cultural pressure exerted by their 2nd city that otherwise needs a monument to slowly accumulate culture - this lets them claim a ridiculous amount of land and resources simply with a single tech. Finally, the AI bonuses from immortal to deity are up to
four times greater than the previous incremental increases. Units are 85% as expensive on emperor, 80% on immortal, and
60% the cost for the player on deity. Combined with the per-era bonus of teching, production, and growth bonuses all increasing 1% for the AI, and it makes the computer players both fast and deadly.
These bonuses essentially put you on a strict timer, where if you get too far into the game without dealing with the strongest opponents, you're basically left hopelessly in the dust with no way to catch up against someone far stronger and more advanced than you in every way (coincidentally, the fact that the later civ games lack this sense of urgency, is one of the main reasons why they're bad - you can be in the information era in V and have 5 cities to Shaka's 25 and still roll him over laughably easily with stealth bombers + XCom, because you can defeat 100 units with 10 no problem, regardless of what difficulty). As an example of the increased pace, AI on emperor usually get lib by about 1400-1500AD, on immortal about 1000AD, but on deity as early as 600AD on quite a few maps (I've seen Hannibal nab it 175AD on my playthrough of the cooked Charlie map played by
@Lain about a year ago).
2. is still a very big jump - deity level players will roll over any AI emperor or below regardless, but for newbies, monarch is when the game effectively takes the "kiddie gloves" off. Archery for all AIs means you can no longer warrior-rush them (and barbs are far greater of a threat), and the bonuses they get are "twice" that which they receive on prince (-10% costs compared to -5%), which is usually when you begin to see enough of a difference to the point where you can no longer win without knowing basic game mechanics like chopping, cottaging, and whipping, as well as when AIs can train actual stacks instead of pathetic 6-unit armies, and tech noticeably faster.
Finally, 3. is less of a jump than the other two but still visible. First, on immortal, AIs start with a worker + agriculture. This is huge, as it allows them to start improving food and growing cities immediately, instead of waiting 50 turns to research the proper worker techs like religious-flavor AIs tend to do. But the bonuses accrued on this level...I dunno, from a gameplay perspective,
also hit a "sweet spot" where the size of their armies and speed of teching/expansion jumps up noticeably from emperor. There's a reason that on this level, as stated before, AIs start being serious contenders for lib (especially if you have a bad start/crash your economy), where they can field stacks that can threaten to lose you the game if you're not careful, where they reach feudalism/engineering early enough to stop you from easily running over everyone with HAs/elepults, and where their army size and tech is good enough to tip the scales in favor of libbing cuirs instead of pre-Renaissance units to do a breakout.