I Renamed all variables named after KeyWords and removed duplicates. Also, I hate overloading Event Subs, so I put the working code into its own Procedure.
Option Explicit

Private Sub CommandButton1_Click()
    MoveDataToFinishedTable
End Sub

Private Sub MoveDataToFinishedTable()
Dim Cel As Range
Dim Src As Worksheet
Dim Trgt As Worksheet
Dim Rw As Long
Dim Lr As Long

    Set Src = ActiveWorkbook.Worksheets("Raw Table")
    Set Trgt = ActiveWorkbook.Worksheets("Finished Table")
    With Src 'Find Required Rows
        Lr = WorksheetFunction.Max(.Cells(Rows.Count, "K").End(xlUp).Row, _
                 .Cells(Rows.Count, "M").End(xlUp).Row, _
                 .Cells(Rows.Count, "O").End(xlUp).Row)
    
    ' Start copying to row 4 in Trgt sheet
    Rw = 4

Application.ScreenUpdating = False

For Each Cel In .Range("K4:K" & Lr)    'Do Required rows
    If Cel = "Well being course" Then
       ' to only copy A:E,K:L,S:U
        .Rows(Cel.Row).Union(Range("A:E"), Range("K:L"), Range("S:U")).Copy
        Trgt.Cells(Rw, "A").PasteSpecial Paste:=xlPasteValues
           'to paste into Cols A:Rw
        Rw = Rw + 1
   End If
Next Cel

For Each Cel In .Range("M4:M" & Lr)    'Do Required rows
    If Cel = "Celatherapy" Then
        ' to only copy A:E, M:N, S:U
        .Rows(Cel.Row).Union(Range("A:E"), Range("M:N"), Range("S:U")).Copy
        Trgt.Cells(Rw, "A").PasteSpecial Paste:=xlPasteValues
        'to paste into Cols A:Rw
        Rw = Rw + 1
   End If
Next Cel

For Each Cel In .Range("O4:O" & Lr)    'Do Required rows
    If Cel = "Celitation" Then
        ' to only copy A:E, O:P, S:U
       .Rows(Cel.Row).Union(Range("A:E"), Range("O:P"), Range("S:U")).Copy
        Trgt.Cells(Rw, "A").PasteSpecial Paste:=xlPasteValues
        'to paste into Cols A:Rw
        Rw = Rw + 1
   End If
Next Cel

 End With 'Src

Application.ScreenUpdating = True
End Sub