For me they are not the same, as it is not the same Cano and Caño, but in practical terms it is pretty common in the US that the tilde is lost. It's up to you. I don't know if they would issue a US passport with tilde or accents. I'm not sure if they would do this even in other countries. Does your country of origin passport include the tilde both in the biographic data and on the machine readable section at the bottom of the photo page? I'm not sure whether the machine readable section admits any kind of symbols on top of characters that includes ^öóñ and others.
How is it currently spelled in your Green Card? I mean, ñ or n?