There are a variety of ways you could create your report. To me, the simplest would seem to be a "Joined Report". I say that because the previous calendar year's data won't be changing. Instead, only the current year's data is being updated. Essentially, you'd be creating the same report with two fiscal calendar years, joining them together (duplicating them actually & just changing the date ranges) then updating the report as the current year progresses. I highly recommend you go through the Salesforce1 Reporting Workbook paying particular attention to Tutorial 4 which covers Joined Reports and Tutorial 3 which covers Matrix Reports.
For Year to Year comparisons on a week by week basis, enabling History Tracking wouldn't be very helpful as it only holds a limited amount of data, usually the previous 3 months plus the current month (unless you create a custom object to store additional historical data). Currently, you don't have that data stored to work with.
To elaborate a bit, with History Tracking enabled, you could Set Up Historical Trend Reporting. See TUTORIAL 5: CREATING A PIPELINE PUSH REPORT in the Salesforce1 Reporting Workbook plus another document called REPORTING ON HISTORICAL CHANGES IN YOUR DATA which your Non Profit might find helpful for tracking its quarterly fundraising progress. That said, I don't believe that's the type of report which would satisfy your current reporting needs for donations and pledges on a year to year basis (called opportunities in regular SF language) and would recommend either a Joined Report or Matrix Report along the lines of what @Fifedog has suggested instead.
@Fifedog has a very novel approach to creating a Matrix Report that may work quite well for you. You may be able to make it satisfy your specific needs as it would line things up very close to what you've shown us. In any case, I think you'll discover that the documentation I've referenced will likely make things easier for you.