Interesting how the folks who love Illuminations reference "goosebumps" and "tears."
To me, this is no contest. Illuminations is the best 13 minutes Disney has to offer. I can do trips without seeing Fantasmic, but I absolutely must see Illuminations.
I agree that for first timers, I would recommend seeing Fantasmic first because of the characters, the water screens, and other novelty technologies that blow away the uninitiated. But RoE offers a show on a deeper emotional level, and the music is a phenomenal and masterfully composed piece.
Then there's the RoE Holiday version. There is not a more moving and powerful 5 minutes anywhere.