First jpg does not support transparency and therefore the white background.
Second for a long time the edge would be trimmed when using save for web. (or is it the export as?) Anyway, I would assume it was added back in because of the complaints that some needed an easy way to center the image.
Lastly that may be one difference between save for web and export as - Now I am going to have to experiment.
I will quick add if you want it transparent on the edge, you would need a format like png.
my design doesnt have any transparency (you can see in the second img) i dont know why the save for web or export as add that white space or transparency.
and the screen on the first img is save for web
Ah, sorry I somehow missed that. I know artboards can play havoc on a layout when they are not really needed. Try ungrouping the artboard and see if that helps. Save first, just in case it have the opposite effect.