Resemblance of molecular orbital formed by mixing of s and p orbitals to hybridised sp orbital
It's important to realize that both of these drawings are highly stylized renderings of the electron density that are intended to provide a qualitative understanding. They don't accurately represent the actual shape of the orbitals. For example, $p$ orbitals are much rounder and taller relative to their length than we usually draw them. They are drawn elongated so that they don't cover each other up if you are drawing more than one on an atom. So the fact that the drawings of the $sp$ hybrid and the $p+s$ $\sigma$ bond look similar should only be taken to mean that they both have a bias of density to one side. You can't really conclude much else from the drawings. 

The Orbitron website is a great resource for seeing the actual shape of the electron density of different types of atomic orbitals, which you can compare to the drawings. 

MolCalc produces similar images of molecular-level orbitals. The bonding orbital of HF is probably the closest you can get to the $p+s$ $\sigma$ bond orbital in your drawing.