Ok so in this situation are we born the opposite gender or do we somehow become the opposite gender now? Because that would really change my answer.
If I was born a female, I can't really answer how I'd be, since so much of who I am today can be traced back to my masculinity, even the little stuff. I might not be that different, just with even longer hair (my hair already goes to my shoulders, I'd imagine it would run the full length of my back as a girl), a larger bust, still into videogames because I grew up in the country with nothing else to do, etc. Or I could be completely different, there's no way for me to tell. Societal pressures would be far different, I'd be looked at as even more unusual due to my affinity for mathematics and the sciences, and so on. I can't really answer that one wholeheartedly.
But if I was gender-swapped tomorrow, then I can give a better answer. I'd feel very, very weird. Knowing that inside I am actually a male, I would talk to people much differently. Since I'm assuming everybody else would think I'd always been a girl, I'd have to be more cautious about how I talk to other girls. My natural attraction to girls would still be there, so I'd be looked at as a lesbian. I'd be very freaked out by all the female-specific body issues I'd have to deal with. I'd juggle my jugs for a minute or two, just being honest, then get over it. It might be interesting to see the world from the perspective of females for a short while, but if the change was permanent I'd be doing everything in my power to return to my male body; it would drive me crazy to be forced into a gender I wasn't born into.