According to my calculations in sage, 13 of the 156 graphs on six vertices have simple Laplacian eigenvalues but repeated adjacency eigenvalues. The first of the 13 is a tree with adjacency matrix: $$\left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{array}\right)$$ There are no examples on less than six vertices.
Note that if $M$ is symmetric and $D$ is diagonal, there is essentially no relation between the eigenvalues of $M$ and the eigenvalues of $M+D$ and so it would be very surprising if there was a relation of the sort you asked for.
According to my calculations in sage, 13 of the 156 graphs on six vertices have simple Laplacian eigenvalues but repeated adjacency eigenvalues. The first of the 13 is a tree with adjacency matrix: $$\left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{array}\right)$$