Downsampling and Gradient filtering
As the commenters have stated, this is not the same. The relationships between down/up-sampling and filtering are formally described by the noble identities.

In your case down-sampling followed by filtering with [-1 0 1] is equivalent to filtering with [-1 0 0 0 1] first and than downsampling.

In any case you need to make sure your don't alias before down-sampling the picture.