I'm trying to understand the difference between Upper Triangular and Diagonal Matrices for square matrices. There is a theorem that says every matrix can be made into an upper triangular matrix. Now if my matrix as the same number of eigenvectors as its dimension then I can make a Diagonal matrix by changing the basis to the eigenspace. So in this case, why will I ever prefer to make a matrix Upper Triangular(and not diagonal) when I can make it diagonal?
Perhaps that you are missing the fact that every diagonal matrix is upper triangular too. So, if a matrix is diagonalizable, it is, by definition, similar to a diagonal matrix. Otherwise, it is not similar to a diagonal matrix, but it is still similar to an upper triangular one (at least over an algebraically closed field, such as $\mathbb C$).