Matt is correct about the firm name.
In both your examples, it looks like the input data is consistent, with the first line being more precise than the second line. If that is always the case with your data then you are golden. Stick with the first line as the recipient (or firm name) and the second line as "extra data". That "extra data" is really irrelevant from an address validation perspective since the address would get to the location regardless of the "extra data". The USPS relies on the address data more than "referential data" (except in the case of the North Pole, which we all know has only one valid address).
I took the liberty of submitting some variations of your addresses to be validated against the USPS data. I was looking to see if the USPS had the Firm Name attached to the address in either of these two cases. Nope.
barnes & noble
union square
33 e 17th st
10003
barnes & noble union square
33 e 17th st
10003
barnes & noble
33 e 17th st
10003
union square barnes & noble
33 e 17th st
10003
Koi Restaurant
Bryant Park Hotel
40 W 40th St
10018
Bryant Park Hotel
Koi Restaurant
40 W 40th St
10018
Bryant Park Hotel Koi Restaurant
40 W 40th St
10018
In each case the address was parsed out correctly while the addressee AND the "extra data" was ignored. I hope that gives a little insight into how the USPS address validation process works.