Wake was within the planned Japanese defensive perimeter in the Pacific, and was also close enough to the Marshall Islands to be used as a base to bomb them (Japan attacked Wake from the Marshall Islands, using G3m (Betty) bombers, destroying 8 of the 12 Wildcat fighters on Wake on the ground, so the possibility of raids the other would have been obvious to them). Once captured, Wake Island was garrisoned and fortified by the Japanese, with the garrison reaching about 4100 men in January 1944 when it was reinforced as a US attack was expected. Repeated bombing attacks killed about 600 of the Japanese defenders, and about 1300 died due to starvation and disease. The remaining half of the garrison surrendered on the 4th of September 1945.
The problem that the Japanese had was that they needed to defend all islands within reach of the US forces within their defensive perimeter, or they'd risk giving the US free bases they didn't have to fight for. The US, on the other hand, could pick and choose where to attack. The US chose to push their central Pacific drive through the Marshalls and Marianas (which required attacking Tarawa as the first step). Essentially, the Gilberts and Marshalls provided a better chain of bases to support the central Pacific drive than the rather isolated Wake. This meant that Wake Island could be largely ignored, and the garrison there left to starve.
Note that the Japanese garrison on Wake was similar in size to those on other islands near the Japanese defensive perimeter, such as Tarawa, Kwajalein, and Eniwetok.
A map showing the Japanese defensive zone, Wake Island, and the path of the central Pacific drive:
