You're correct, not much difference between Cicero or Clay. A few minor differences:
-Parts of Clay have lower taxes than Cicero.
-Since the Cicero Swamp is in Cicero... Clay may have fewer mosquitoes
-Some parts of Cicero near the swamp have basement flooding issues in rainy Springs, but a house with a sump pump usually solves that problem
-IMO the best part of the CNS school district is in the Clay neighborhoods surrounding the Bear Road/Buckley Road intersection....Wild Creek, some of developments just west of the Village of North Syracuse all the near Wegmans on Taft Road. Homes there keep there value and are always in high demand.
Lots of homes have pools in the Cicero/Clay area. Mostly the uglier above ground pools. Many neighbors at one time or another went through a phase of buying a pool, then realizing they are not worth the cost or effort to maintain...then they pull them out. I've never had one myself... never much of swimmer.
Hope this helps! Good Luck!