This does seem slower than I would expect. You could try this version using ranges but it doesn't show a significant speed improvement.
Code:
Sub Change_Value_to_Next_Cell_GOOD()
Dim aRng As Range
Dim sText As String
sText = InputBox("Enter text to move cell")
Set aRng = ActiveDocument.Range
ActiveWindow.View = wdNormalView
Application.ScreenUpdating = False
With aRng.Find
.ClearFormatting
.Text = sText
.Wrap = wdFindStop
Do While .Execute
If aRng.Information(wdWithInTable) Then
aRng.Cells(1).Next.Range.FormattedText = aRng.FormattedText
aRng.Delete
aRng.MoveStart Unit:=wdCell, Count:=2
End If
Loop
End With
Application.ScreenUpdating = True
MsgBox " You're Selection is Done in the Active Document!"
End Sub