Short answer

Yes in this particular case both do the same. Removing Range("A1") is fine.

Long Answer

This is due to the fact you are using ActiveCell in this line:

ActiveCell.Offset(1, 0).Range("A1").Select

The ActiveCell is the first cell within the range you have selected.

Consider the following macro:

Sub Macro1()
    Debug.Print ActiveCell.Address
End Sub

Whatever range you select this will print the address of the white cell within the selection.

i.e.

The ActiveCell is $A$4

Example

Calling Offset(1,0) on a single cell will only offset that cell. So if we look at your original code:

Sub Macro2()
    Sheets("Setup").Select
    Range("Start").Select
    ActiveCell.Offset(1, 0).Range("A1").Select
End Sub

Let us assume that my previous selection shown (A2:B4) is the named range of "Start" and we can walk through exactly what is happening:

1. In this example Range("Start").Select would select range A2:B4. Thus making ActiveCell equal to A2.

2. Next we call Offset(1,0) on ActiveCell which is equivalent to Range("A2").Offset(1,0) putting us at range A3 (1 row below A2)

3. Now we call .Range("A1") which is going to grab the first cell within the range. Since the current range is only A3, .Range("A1") gives us A3.

4. Then of course .Select() is still only selecting A3

When is .Range("A1") actually useful?

Consider the following example without any Range("A1") call:

Sub Macro3()
    Sheets("Setup").Select
    Range("Start").Select
    Selection.Offset(1, 0).Select
End Sub

Since we have changed ActiveCell to Selection the Offset(1,0) will select the same dimension range as "Start" just offset-ed by 1 row.

i.e.:

If this is the range of "Start":

We run the example macro:

We have a new selection of the same dimension.

However if we change the example macro to include the Range("A1"):

Sub Macro4()
    Sheets("Setup").Select
    Range("Start").Select
    Selection.Offset(1, 0).Range("A1").Select
End Sub

Only the first cell in the selection is now selected.