Convert your datemodified field into a date without time and do the same to getdate(), then you can compare them...
CONVERT(NVARCHAR(50),datemodified,103) = CONVERT(NVARCHAR(50),GETDATE(),103)
so you get:
SELECT Assignment, Datemodified, General, IncNumber, NextSteps, PDCRStatus, RootCause, Status, Summary, Timings, UserID FROM Turnover WHERE DATEPART(HOUR, datemodified) between 06 and 18 AND CONVERT(NVARCHAR(50),datemodified,103) = CONVERT(NVARCHAR(50),GETDATE(),103)