Remove Rows and Columns in a matrix

I don't think you're using Delete correctly. For comparison:

remover[
  m_?MatrixQ,
  rows : {___Integer}, cols : {___Integer}] :=
 Part[
  Delete[m, List /@ rows],
  All, Complement[Range@Length@m[[1]], cols]]

removerr[m_, rows_, cols_] :=
 Module[{r, c},
  r = Complement[Range@Length@m, rows];
  c = Complement[Range@Length@First@m, cols];
  m[[r]][[All, c]]]

SeedRandom[1];

test = RandomInteger[{1, 10}, {5, 3}];

{remove[test, {1, 2, 5}, {2}], removerr[test, {1, 2, 5}, {2}]}

{{{9, 1}, {5, 9}}, {{9, 1}, {5, 9}}}

You need to use UnSameQ instead of Unequal. As Null does not work with ==.

Your function also deleted a row twice.

ClearAll[RemoveRowsColumns];

RemoveRowsColumns[expr_, a_, b_: Null] :=

   Module[{M},

       M = Delete[expr, a];

       If[
          b =!= Null,

          M = Transpose[Delete[Transpose[M], b]],

          M
        ];
      M
  ]

Refactered more

ClearAll[RemoveRowsColumns];
RemoveRowsColumns[expr_, row_, col_: Null] := 

    Module[

        {M = expr},


        If[col === Null,

            M = Delete[M, row],

            M = Delete[M, row];
            M = Transpose[Delete[Transpose[M], col]];

        ]; 

    M]

Note the special syntax when you call it with lists of rows and columns: You have to wrap the row/column numbers in { }(due to Delete):

I will leave it to you to modify RemoveRowsColumns to negate the need for this.

 SeedRandom[1];
(test = RandomInteger[{1, 10}, {5, 3}]) // MatrixForm
RemoveRowsColumns[test, {{1}, {2}, {5}}, {{2}}] // MatrixForm

{{9,1},{5,9}}

The If-statement can be eliminated by choosing the default argument to fit into the same code as a list of parts:

MatrixForm[test = RandomInteger[{0, 9}, {5, 8}]]

RemoveRowsColumns[expr_, a_: {}, b_: {}] := Transpose[Delete[
     Transpose[Delete[expr, Transpose[{a}]]], Transpose[{b}]]]

MatrixForm[RemoveRowsColumns[test, {1, 3, 4}, {3, 4, 5}]]