How does the Modified Phong Lighting Model from the Phong Lighting Model?
The Phong lighting model computes the specular response as the dot product between the mirror reflection direction and the viewing direction, raised to a power.

For example, if $\vec{V}$ is the viewing direction, $\vec{L}$ the incoming light direction and $\vec{R}$ the perfect specular reflection direction for $\vec{L}$, then the specular response is $\text{max}(\vec{V}.\vec{R},0)^p$, where $p$ is the exponent that controls the fall-off of the specular response. 

The Blinn-Phong model uses a half vector $\vec{H}$, which is computed as $\vec{H} = \frac{\vec{L}+\vec{V}}{|\vec{L}+\vec{V}|}$, which is then used to compute the specular response as $\text{max}(\vec{H}.\vec{N},0)^p$, where $\vec{N}$ is the surface normal. The idea is that when the eye vector $\vec{V}$ is aligned perfectly with the perfect mirror direction $\vec{R}$, the half vector $\vec{H}$ would be exactly aligned with the surface normal $\vec{N}$. Hence the Blinn-Phong can get away with computing an expensive mirror direction $\vec{R}$ everytime by using the half vector which can model the specular response.