What Grade Jumps Have The Biggest Quality Difference?

AU58 versus MS60 is probably the biggest difference, but it's in the opposite direction - an AU58 coin will typically have much better quality than a MS60 coin. (Yet it's usually the latter that costs more.)
I don't think any MS grades are really that much different from each other - but in the higher end especially. MS67/68/69/70 is in the eye of the beholder, and 66 isn't far behind either.

I'm not sure which specific grades are or are not used in the middle of the scale (I know 5 and 59 don't exist, but I'm more fuzzy on the rest), so I can't claim a specific jump as the biggest aside from the AU58/MS60 anti-jump; AG3/G4 and G6/VG8 (if that's a single step) are major candidates, but there's probably something bigger in the vicinity of VF.

I would have to say the difference between AU-58 and MS-60.
Where the 58 shows wear and the 60 does not .

It is absolutely impossible to "blanket answer" your question.

The assumption that all coins in like grade sell for nearly the same price is the basis for the flaw in that thinking.

Check out sold prices with either PCGS or NGC on Heritage, for example, and you can find 100%+ differences on the "same" coin.

While the grade may be the same there can be huge differences in the same coin in the same grade because of its eye appeal.

Try finding a pretty MS-67 (not proof) Franklin half and the average price will be blown out of water.

I think I need to clarify - I'm not saying MS61 and MS69 are the same thing, I'm just saying that it's a pretty narrow range and most of it is concentrated in the lower end of the numbers.
MS60/61 is probably the biggest jump, since the 60 would include a (relatively) wide range of "technically uncirculated". (Then again, beyond some point they'd just get Details grades, so maybe not; some googling indicates that a straight MS60 grade is actually very rare - I couldn't find a single one on CCF.)
MS61/62 is not a big gap, but only because there aren't very many MS61 coins in the first place (actually I'm not sure that grade exists at all); 62/63/64/65 are all relatively significant jumps (and which one is bigger probably depends on the series).
Beyond that the grades get rather nebulous, and 68/69/70 is definitely a case of being less or more lucky.

As has been posted .. pretty much impossible to answer this question.

You can have two coins, say Morgan dollars .. same grade and one looks
better than the other.

You can take ten different collectors, ask them all to grade the same coin.
Most often you will get two or three difference of opinion on grade, for those
ten collectors.