The difference between these terms seems to be, in practice, largely one of semantics, and there is little discussion in contemporary sources on the difference between them. A few sources have diallektes and aesymnetes as synonyms. For example, the (German - translated) Wikipedia on aesymnetes states:
An Aisymnet (ancient Greek αἰσυμνητήρ aisymnetér, also Diallaktes
διαλλακτής ) was originally a " referee " in ancient Greece of the 7th
and 6th centuries BC .
The English aesymnetes page, though, makes no mention of diallaktes:
Aesymnetes (Greek: αἰσυμνήτης, from αἶσα, aisa, a "just portion",
hence "a person who gives everyone their just portion") was the name
of an ancient Greek elected office similar to, and sometimes
indistinguishable from, tyrant.
Aristotle defined aesymnetes as 'an elected tyrant' and applied the term to Pittacus, a general from Mytilene who held the position for ten years and then resigned. Michael Grant, in The Rise of the Greeks, defines aesymnetes as 'arbitrator' or 'umpire', while J.B. Bury also assigns 'legislator' to the term.
The most commonly given definition for diallaktes is 'mediator' or sometimes 'arbiter' (though the latter is given in the context of diplomacy between Greek states). Diallaktes is most commonly used when referring to Solon of Athens, and this was the term both Aristotle and Plutarch used for him. Solon himself refused the role of tyrant, but both he and Pittacus were chosen to resolve differences between opposing factions in their respective cities and both gave up their roles voluntarily.
One possible difference, though, emerges in Plutarch where it is implied that, initially, Solon's position of diallaktes did not include the power to revise laws or the constitution; these powers were only given to him later. It is thus possible that the position of diallaktes wasn't as all-embracing in terms of power as that of aesymnetes, in which case one could argue that, although Solon started out as a diallaktes, he ended up as an aesymnetes.
It should be noted that the word 'tyrant' did not necessarily have negative connotations to ancient Greeks until the late sixth century - a tyrant could be good or bad, depending on how he used his power.
J. H. Blok & A. P. M. H. Lardinois (eds), Solon of Athens
Victor Ehrenberg, From Solon to Socrates
Kelcy Shannon Sagstetter, Solon of Athens: The Man, the Myth, the Tyrant? (dissertation)
Anna-Lena Svensson-McCarthy, The International Law of Human Rights and States of Exception