Why rear derailer has limit on front chainring difference? Just as an example, Shimano RD-M786-SGS (as most other RDs) specification says &quot;Max. front difference: 22T&quot;.
Why is rear derailler limited in front max difference? It is clear that it is limited in total tooth capacity, but max front difference is stated separately. What goes wrong if it is exceeded, given that other numbers remain within their specified limits?
It is not directly connected to the difference between the front chainrings.
Instead, a Rear Derailleur will have a &quot;capacity&quot; which is the total amount of chain it can take up before bottoming out.
The RD also has a value for the small cog (often 11 tooth) and either a maximum-sized cog, or some of the new specs give a maximum and a minimum tooth count for the big cog.
For example, that specs page linked in your question says

Low sprocket_Max. 36T 
Low sprocket_Min. 32T 
Top sprocket_Max and Min. 11T 
Total capacity    43T 
Max. front difference 22T

The cassette can &quot;eat up&quot; between 21 and 25 links, and the total capacity of the RD is 43T.
43T less the 21T consumed by the smallest allowed big-cog is 22 T
So your bike might have a 48T big chainring, and the smallest chainring is 26T.  Any middle chainring has to be between those.
You could have a 53 tooth big chainring, but your smallest could be no smaller than 31 tooth.
summary  The chainring difference is easier to understand for consumers, its just processed the other values.

Putting this another way: Imagine your hardest gear is 48:11 and easiest gearing is 26:32.  The rear mech will be almost flat, pointing forward and the chain through will be near straight.
Changing chainrings releases ~22 links of chain, which adds to slack.  Your rear mech has to fold up and take up ~22 links, now its in 26:11.
Then changing up the cassette uses 32-11 = 21 links of chain.  Thats the lower boundry.  If you had a 36T max on the rear, it would use up 36-11 = 25 links of chain.  Those two values of 21 and 25 straddle the 22 tooth number quoted in the spec.
Note that Shimano's specs are notoriously conservative too, you can almost always add 2 teeth to any maximum and still have perfectly good performance.

If you had far more difference between your chainrings than 22T, it wouldn't shift very well.  16T is quoted as a reasonable maximum between adjacent chainrings, and 22T would imply a triple crankset with a third chainring.
So if you exceeded the front difference of 22T with (say) 30T, then moving from big chainring to small would release ~30 links of chain, the extra 8 links has to go somewhere.  This might let the derailleur rub the chain back on itself in a small-small gearing, or would not put enough tension on the chain resulting in sloppyness and possibly chain slip.
Example: This chain is too long, and is rubbing on itself.  The gear is a small:small combo so there is the most &quot;spare&quot; chain outside of the chainrings/cassette.
  From Is my chain too long?