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Not really, it had multiple internal changes to permit easier/cheaper production, higher rpm and increased boost pressures. You could call it a complete rework of the DB 601.DB605 was not a separate engine program. It's essentially just a newer version of the DB601.
As a second question, I wonder if anyone knows why the DB 603AM or any other methanol cooling system on the DB 603 didn't appear. The Jumo 213 was able to get it. IIRC it took until the DB 603LA (for Ta 152C) before it became standard for the DB 603 series. But those were of course never produced in any large numbers.
Dessau 1086 prototypes, test engines
Magdeburg 24267 July 1937 to Aug. 1944
Kothen 20911 1938 to Feb. 1944
Leipzig 17032 1942 to Aug. 1944
Stettin 4714 only 1942
Strassburg 238 1943 to Feb. 1944
Focke Wulf tested 3 prototypes to assess performance of Fw 190 with DB 603. Fw 190 V-13, V-15 and V-16. The last one flew in August 1942. There were no major problems with the installation and in a few months production could have been started. I have created a graph comparing Fw 190 A and C performance.