First off, some linear algebra systems are smart enough to only store half of the matrix, this could save you a bunch of memory. But even if this isn't the case, various algorithms in numerical linear algebra will exploit the symmetry.
For example, given a symmetric matrix, any eigensolver will immediately know that all eigenvalues are real-valued, and the solution method may use that fact.
A typical thing that many people will think of are Krylov subspace methods for the solution of equation systems $Ax=b$: If your problem is symmetric, you know that you don't need methods for nonsymmetric problem like GMRES, and can reside to something less memory-intensive like MINRES, or -- if your matrix is also positive-definite -- CG. The convergence behavior of Krylov methods is not influenced by permutations though, so you could even use symmetric methods for your unpermuted system.
Another example is the factorization of your matrix $A=LU$ in to an lower-triangular part $L$ and an upper triangular part $U$. If $A$ is symmetric, then $A=LL^T$, and you only have to store one factor (Cholesky decomposition).