As noted in other answers but salient: ATCs handle multiple - sometimes dozens - of aircraft simultaneously while pilots operate a single aircraft. A disastrous error by a pilot can kill hundreds but even in a mid-air between two fully loaded 747s, there's two pilots involved and responsibility is distributed. A similarly severe ATC error can have double the consequences, or more.
Pilots operate machines, while ATCs do their best to influence (thankfully quite cooperative) human behavior. At the end of the day, the NAS is chaotic in ways that flight just isn't. Pilots receive direct, observable feedback to their command/control inputs to aircraft operating normally. ATC, especially in TRACON and En-Route positions, receive delayed, probabilistic feedback for the actions they take. The anticipatory tension experienced by ATCs is thus higher. Pilots do have more skin in the game - they're on board their craft - but they have more immediate control over what situations their craft experiences as well.
ATCs, consequently, suffer elevated health consequences compared to pilots.
These include:
From this 1978 study - Hypertension (substantially elevated risk), respiratory illness (not super conclusive because of wide variance in diagnostic criteria), peptic ulcers
From this earlier study - Hypertension, Peptic Ulcers, Diabetes
From a 2015 study - Poor sleep quality, poor quality of life scores, concentration issues, higher body fat percentages with all comorbidities that implies.
The selection of age 56 specifically was not scientifically determined but is consistent with the maximum hiring age of 30 allowing 25 years of service, sufficient to justify pension benefits and other retirement perks while still being earlier enough in a retired controller's life as to allow a second career in less stressful work.
It's important to acknowledge that studies since the 70s, when this retirement age was laid down in law, don't find the evidence as compelling as was understood to be the case at that time. Nor, however, is it dispositive. More modern evidence suggests that the health impacts of ATC work have as much to do with the work culture, structure of shifts, and staffing shortages as anything else - i.e. it's not the work itself, but how it's being performed.
Evidence does exist, as of 2005 anyway, that controller performance suffers more variability as a function of age in ATCs than in other types of work - and certainly attention to detail and concentration issues - which also correlate with age - are especially deleterious to the work ATCs do.
None of this is perfect, but the studies from the 1970s did use airmen as a comparison/control group.
TL;DR - Pilots and ATCs suffer different kinds of stress which acts on the body and mind in different ways. ATCs are required to process far more discrete information at a constant rate and have far less immediate and direct control over the systems they operate. At the time the mandatory retirement age was instituted, best available scientific research suggested that ATCs burned out more quickly and thus there was merit to making them eligible for retirement benefits earlier and safety reasons to force them to take it. Our understanding of the situation is evolving, but the law is always slower to change than our knowledge.