What is the difference between the climate of salt lake county and Utah county?

None, really. Sometimes it will be raining or snowing in one county and not the other, but their climates are the same.

Another Answer

Only 45 miles separate the county seats of these two counties. But today in Utah county the temperature is 81 degrees F, humidity of 37%. In Salt Lake County, the temperature is 91 degrees F, humidity 20%. The differences are probably attributable to the heavier density of buildings and infrastructure in Salt Lake County.