The S15 tag shows a total that includes all transactions that were not refunded within the campaign specified.

The Donations by Transaction report inherently filters out ticket event purchases and all transactions associated with both TeamRaiser and Personal Fundraising registrations (registration fees, additional gifts, premium purchases).

The Donations by Transaction report also shows information about receipts.
When it includes information about receipts, it includes one row per receipt. 
If a specific transaction has had a receipt regenerated, that transaction will be included multiple times and duplicate values would be included in the total if this is not accounted for.

To get information that matches the S15 tag, run the following report instead:

Data Management > Reports > Report Writer (tab) > Create New Report (button) > Choose Transactions (dropdown list) > Choose Transaction detail (radio button)

Choose the net value transacted field plus additional fields from Biographical Information and Valued Transaction sections.  You may be able to include fields from other sections as well however do not choose fields from the receipt section, otherwise it will introduce the same issue as we saw in the donations by transaction report in which regenerated receipts will cause multiple rows to appear for the same transaction.

Filter for the campaign via the campaign_id field.
Filter on transaction purpose for debit only (so you don't get declines and refunds).

Export the results and use a spreadsheet program to sum the results for the net value transacted field.