I'll begin my saying I use neither product, but what it really boils down to is what kind of camera you'll be using. Since you said your only camera is the iPhone (not sure which model), then Aperture may be better suited for you. Another major factor is the cost. Aperture is $79.99 while Lightroom is $299. Lightroom, however, is MUCH more powerful when it comes down to editing photos and has many more advanced feature more akin to Photoshop. Aperture is a bit more simple to use, but not as powerful in making edits or changes to photos. Both can be used for basic photo editing though, so it just depends on the expectations you have as far as that goes.
When it comes to organization, good things can be said for either program, and the lower cost of Aperture may be an advantage for you. If you're only going to use the software for the sole purpose of organizing your photos, then Picasa might be the way to go, however may Pro and Semi-Pro photographers use Lightroom for organization, especially with the more advanced photo editing features. Many people also believe that Lightroom works better with RAW file types, which aren't supported by the iPhone anyway. iPhone is JPEG only.
It could also make a bit of a difference if you're using a PC vs using a Mac. If all you need is photo organization and you have a Mac, then iPhoto should be all you'll need to keep things organized. For a PC, Picasa (even though I'm not a fan of Google) may be a good choice too.
You should probably also do some additional research about both programs to see which one you feel might be a better suit. If I had to recommend software for someone who only uses an iPhone for pictures, my personal opinion is that Lightroom would be overkill.
Hope I've helped!