Soil temperature can have an affect on grass growth which is what causes the grass to become green. If your neighbors apply synthetic nutrients, less dependant on the Soil Food Web, their lawns can green up quicker, but will require more poisons to control insect pests that will be atrracted to those sick lawns. Every lawn I have seen that uses the synthetic growing stimulants also requires numerous and expensive application of poisons to control the insect pests that are attracted there and that contributes to the poisoning of our planet, the world we live in.
Thomas Berry said it best, "You cannot have healthy people in a sick planet."

Various varieties do green up quicker than other varieties, even in the same lawn. I wouldn't worry about whose lawn comes in first, but I wonder why you're fertilizing dormant grass.
Better to wait a while after mowing for cool-season lawns. :)

I started organic last year with some scotts organic and some milorganite. It takes a little while in spring until the lawn wakes up and the microbes digest the milorganite. Organic fertilizer is slow release unlike chemicals. Give it some time

I put soybean meal down, about 50 lbs and got good results.I used alfalfa too but I think the sb meal really greens up a yard. I live in FL so a bit different soil, but give it a shot.