I might be having a brain fart here but from these two definitions, I actually can't tell the difference between a complete graph and a simple graph.

I might be having a brain fart here but from these two definitions, I actually can't tell the difference between a complete graph and a simple graph.

On
A simple graph is a graph that does not contain any loops or parallel edges. So, the vertex $u$ is not adjacent to itself and if the vertex $u$ is adjacent to the vertex $v$, then there exists only one edge $uv$. A complete graph of order $n$ is a simple graph where every vertex has degree $n-1$. In other words, every vertex in a complete graph is adjacent to every other vertex.
Every complete graph is also a simple graph. However, between any two distinct vertices of a complete graph, there is always exactly one edge; between any two distinct vertices of a simple graph, there is always at most one edge.