Why is there a bigger difference in temperature between January and July in Spokane than in Seattle?

Because Spokane is located in the west of the state of Washington, near the border with Idaho, at an elevation of 1,843 ft, while Seattle is a coastal city.

This makes the climate of Spokane to be classified as a dry summer continental climate, while the climate of Seattle is oceanic.

Continental climates are those located away from the coasts, and tend to be more extreme: hotter summers and colder winters. The climates of coastal areas like oceanic or mediterranean climates, tend to be milder, because of the moderating effect of the ocean.