I've talked to a few people who are uncomfortable riding in the drops. One of them says she feels like she's going to scrape her face on the ground because it's so low.

The position certainly isn't comfortable, but it gets better. As dutret said, there's really only a few times you'd want to ride in the drops, and those are usually for speed or to get out of the wind.

Bullhorns allow you to get a little further out, helping get more aerodynamic, and also give you something nice to grab onto for climbs.