open your saved game with any text editor and look for the line that says difficulty: around line 11, change the number to anything from 0-4 (beginner-superhuman)
here's a few examples of how difficulty comes into play:
on the geoscape:
after (14 - difficulty) months, aliens will start initiating retaliation missions proactively at the end of each month, instead of just reactively when ufos are shot down.
in dogfights, UFOs will break off after a random amount of time adjusted by difficulty, will shoot at a rate dictated by difficulty, and will generate a retaliation mission based on some percentages based on difficulty.
the funding nations will judge you more harshly.
when starting a new battlescape:
enemy unit numbers are based on the difficulty, with beginner/experienced being "low", superhuman being "high" and veteran/genius being a random number between the two.
enemy stats gain an average rate of an extra 4% per difficulty level over beginner, and on beginner, enemy armour values and the firing accuracy stat are halved.
during the battlescape game:
the likelihood of psionic attack ranges from a 25% to a 50% chance.
the likelihood of grenade attack ranges from a 20% to a 33% chance.
explosive efficacy takes difficulty into account when weighing it's decision.
alien blaster launchers are restricted to a certain number of waypoints relative to difficulty (6-14).
as a result of this enquiry, a code cleanup has occurred.