What is the difference between Save For Web vs Exporting?

I want to fully understand what is the difference between these two. What are they used for and their best results? I have created a logo for a photography friend which has subtle details because I masked two different brushes together if that makes sense.

So it is time to get the final and I exported as PNG then I've seen a tutorial from Adobe that the guy saved for web and I did try the same. Exporting was able to show the details but save for web preview wasn't too sharp and the details weren't even there.

P.S. What is a reasonable size for my logo. When I opened the exported file in my browser. It was super up-scaled.