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I can't see this documented in the Esri help files, so I'm entering it here for posterity.

According to Jason Scheirer from the Esri ArcPy team this is the intended behaviour:

The .next() method raises an error by design: it follows to the letter the Python Iterator Protocol, which requires the .next() method on an object to raise a StopIteration exception when it is exhausted.

The suggested behaviour is to use a for iterator on the cursor, ie without using the next() method at all:

with arcpy.da.SearchCursor(fc, fields) as cursor:
    for row in cursor:

In my case I wanted to perform an action when there was no row - I found that a Try, Except block worked well:

with arcpy.da.SearchCursor(fc, fields, where) as cursor:
    try:
        row = cursor.next()
        print(row)
    except StopIteration:
        print("No rows")
    except:
        print("something else went wrong")