There are several reasons why the populations are very different for GPW v3 and GRUMP Alpha. When resampling to lower resolutions (such as one degree) it is important to sum the population counts to get comparable results. There are 546 GPW grid cells in one degree, and 14,400 GRUMP cells. If you do not sum the population the results will grossly underestimate population by very different amounts.
There are differences in the coastlines and extents between the two datasets; GRUMP is based on an earlier version of the GPW boundaries; GPW has a number of improvements. This means that coastal cells could be wholly populated according to GPW and only partially populated in GRUMP, or vice-versa. This will result in large variances.
Finally, GRUMP increases densities in urban extents (lighted grid cells). Even though aggregating to one degree will remove much of this variation, there could be an edge effect where urban cells in GRUMP have much higher population counts than GPW.