Date fields—including calculated fields that returns dates—store and return a data value known as a datetime, which can’t be compared against a string like “5/14/2021”. That’s why your formula is failing. What Airtable does when comparing a datetime against a string is convert the datetime into a raw date string, which looks like this: “2021-05-14T00:00:00.000Z”. Even knowing that, though, this doesn’t make it possible to actually compare datetimes in string format (other than exact equality).
To compare a datetime from a field against a specific date of your choosing, parse your chosen date string using DATETIME_PARSE(), which will return a datetime data item that can be used in datetime comparisons.
That detail that I marked in bold helps a lot. Using the DATETIME_PARSE() function that I mentioned above, this should work:
IF(
{Date Added} >= DATETIME_PARSE("06/01/2021", "MM/DD/YYYY"),
Autonumber,
{Book Number - Old}
)
