Now that makes at least four different answers based on four different understanding of what the OP's question was.
1. File size.
2. Crop factor.
3. Cropping in camera.
4. Changing aspect ratio.
Perhaps if the OP is so disposed, he would re-state his question and pick one of the above answers as the one he was looking for.
There are several factors. The answer is, yes, specifically, it is exactly the same as cropping in the photo editor later. A crop is a crop.
If for example, if shooting cropped APS DX frame size on a full frame 24 megapixel camera...
It is smaller files, and fills the buffer less too.. It gives an apparent telephoto effect (any cropping and subsequent enlargement does that too).
But there are a couple of large downsides.
Instead of 24 megapixels, you only have 10 megapixels. It is a crop, which is a smaller file, and is sometimes adequate (certainly for snapshots on the computer monitor), but it takes getting used to again.
And a 24 mm lens no longer sees a wide 24mm view, but instead sees a cropped 36mm view now. Not even quite a "normal" view now, instead a slightly telephoto view for DX. So you lose wide angle capability, unless you stand back half again farther (assuming 1.5x crop factor).