bump mapping can be equated to an emboss effect. Normal mapping on the other hand actually creates the bump effect based on the light direction. So, yeah, it's an upgraded bump map, but in conjunction with other pixel shaders it can accomplish a lot.

Not to mention the new tech being pixel-displacement maps (or offset maps) which simulates realtime displacement as opposed to just bump mapping, based on your camera's perspective by actually overlapping the pixels in your texture to create perspective.

https://www.reallyslick.com/pictures/offsetmapping.jpg

Not only that, but they can self-shadow which is craaaazy cool.

https://www.delphi3d.net/misc/images/uber_bumps.jpg
a flat plane with an offset map

Very neat stuff.