What is the molecular weight of bilirubin?
The discrepancy is simply due to round-off error.  The atomic masses of these elements are usually given as follows.  These are known as the standard conventional form, or standard formal short form, since they are given without an interval indicating the uncertainty in their values (see  https://en.wikipedia.org/wiki/Standard_atomic_weight):
C = 12.011
H = 1.008
N = 14.007
O = 15.999
Hence 33 C + 36 H + 4 N + 6 O = 584.673
However, if someone were to round off all the masses to four significant figures, but then report the result to five significant figures, we would have:
C = 12.01
H = 1.008
N = 14.01
O = 16.00
And now 33 C + 36 H + 4 N + 6 O = 584.658 = 584.66