I just made a set of Adirondack chairs this summer. First one was white oak, the other three are cypress. I sealed both with a natural oil based stain.
The chairs are only a few months old, but here are my impressions so far:
White oak - looks nicer new, but doesn't weather as well. The oil doesn't penetrate as well and the instructions on the oil say it will need to be recoated two to three times more often than it would need on softwood. If you left them natural it wouldn't be an issue. Even with the stain the color is changing fairly quickly. Also, after the first rain the grain raised quite a bit and I had to resand it, but it has been ok since. Cost would be fairly high if you were buying the lumber - I used it since I had some and couldn't originally find a source for cypress.
Cypress - I found a local source and it was actually fairly cheap - about $40 per chair. They look pretty nice, but the color of the boards seem to vary a lot more than the oak I worked with - from very light brown to almost red. It wasn't as nice to work with though in my opinion - more knots and defects to work around - but might just be the stock I was working with. The chairs are very light which would be a big plus if you plan to move them often. They stain much better than the oak - after a few months they look almost like they day I stained them while the oak already is showing some age. You can also leave the cypress unfinished if you like the silvery grey look. The grain also didn't raise like the oak. I did have one chair get very bad splits in both arms, but again may have just been bad stock. If I build more they will probably be cypress.
Here are pics of them for comparison:
