Make code faster/better best practices

Maybe as attached. The lists of old and new students should be culled for duplicates and trimmed before running.

If you try several methods, I'd be curious as know the timing results.

You could also just use two columns of formulas ...

A couple of changes to ignore case in name comparisons, and to avoid an error if the number of adds or drops is zero:

Sub AddRem(rCur As Range, rOld As Range, rAdd As Range, rDrp As Range)
    Dim nCur    As Long     ' current students
    Dim nOld    As Long     ' prior students

    Dim asAdd() As String   ' array of new
    Dim asDrp() As String   ' array of drops

    Dim iCur    As Long     ' index to current students
    Dim iOld    As Long     ' index to prior students

    Dim nAdd    As Long     ' number of new students
    Dim nDrp    As Long     ' number of drops

    nCur = rCur.Count
    nOld = rOld.Count

    ReDim asAdd(1 To nCur)    ' max  adds is everyone here this year
    ReDim asDrp(1 To nOld)    ' max drops is everyone here last year
    
    ' clear output area
    rAdd.Resize(nCur).ClearContents
    rDrp.Resize(nOld).ClearContents
    
    ' sort both lists
    rCur.Sort Key1:=rCur(1), Order1:=xlAscending, MatchCase:=False, Header:=xlNo
    rOld.Sort Key1:=rOld(1), Order1:=xlAscending, MatchCase:=False, Header:=xlNo

    iCur = 1
    iOld = 1
    Do
        Select Case StrComp(rCur(iCur).Text, rOld(iOld).Text, vbTextCompare)
            Case -1    ' cur < old; current is new
                'rCur(iCur).Select   ' debug
                nAdd = nAdd + 1
                asAdd(nAdd) = rCur(iCur).Text
                iCur = iCur + 1
            
            Case 0  ' cur = old; continuing student
                iCur = iCur + 1
                iOld = iOld + 1
            
            Case 1  ' old < cur; old is a drop
                'rOld(iOld).Select   ' debug
                nDrp = nDrp + 1
                asDrp(nDrp) = rOld(iOld).Text
                iOld = iOld + 1
        End Select
        If iCur > nCur Or iOld > nOld Then Exit Do
    Loop

    ' add any remaining current students to add list
    For iCur = iCur To nCur
        nAdd = nAdd + 1
        asAdd(nAdd) = rCur(iCur).Text
    Next iCur

    ' add any remaining old students to drop list
    For iOld = iOld To nOld
        nDrp = nDrp + 1
        asDrp(nDrp) = rOld(iOld).Text
    Next iOld

    ' output each list if not empty
    If nAdd > 0 Then
        ReDim Preserve asAdd(1 To nAdd)
        rAdd.Resize(nAdd).Value = WorksheetFunction.Transpose(asAdd)
    End If
    
    If nDrp > 0 Then
        ReDim Preserve asDrp(1 To nDrp)
        rDrp.Resize(nDrp).Value = WorksheetFunction.Transpose(asDrp)
    End If
End Sub