I rarely use BETWEEN and 99.x% never use it for dates. I also do not use BirthDate <= '1/1/1980' but rather "<" instead.
The problem with Dates is if they also include Time.
BirthDate BETWEEN '1/1/1970' AND '1/1/1980'
will include anyone with a Date+Time of Midnight on 01-Jan-1980, but not someone with a time of "one minute past midnight"
I've seen people trying to do
BirthDate BETWEEN '1/1/1970' AND '1/1/1980 23:59:59.999'
but that doesn't work because DateTime doesn't hold the time to exactly 3 decimal places ... and DateTime2 has lots MORE decimal places.
I suspect you actually mean "Up to but not including 01-Jan-1980", if so then this would be fine
BirthDate >= '1/1/1970' AND BirthDate < '1/1/1980'
(NOTE "<" and not "<=")
So for dates typically the best test is:
MyDateTimeColumn >= '1/1/1970' AND MyDateTimeColumn < OneUnitAfterTheEndLimit
Lastly, don't use "1/1/1980" for string dates, its ambiguous as to Day / Month. SQL parses string dates based on all sorts of things - including the Selected Language for the Currently Logged on User ... so tomorrow SQL's idea of "1/2/1980" might be 02-Jan, or might be 01-Feb.
The format '19800101" i.e. "yyyymmdd" will always be parsed UNambiguously - so use that format, anything else with day-month-year and any separators / punctuation / spaces is at the mercy of what the parser decides is the current method!
1980-02-01 is no better - try this:
SET LANGUAGE 'us_english'
PRINT 'Test US English'
GO
SELECT CONVERT(datetime, '1980-02-01')
GO
SET LANGUAGE 'french'
PRINT 'Test French'
GO
SELECT CONVERT(datetime, '1980-02-01')
GO