I have not downloaded it, so forgive me if it is there and I don't see it. Still no omnibar? Why a separated dedicated search field?
The main difference is that the Omnibar logs all your keystrokes to Google's servers. The Awesome bar searches through the data it already has locally (bookmarks, history, open tabs), and it makes a Google search if the data you entered isn't a URL. Not logging your keystrokes is a privacy feature. On the other hand, the dedicated search field does, and that's its feature.
I personally can't see the difference in usage between the Omnibar and the Awesome bar, apart from the fact that the search suggestions make more sense in the Awesome bar, since it only uses my past searches. Follow my advice: use the brand new interface customization UX and remove the dedicated search field.
I wish the delete feature were more discoverable, however.
Are the people who's sites you're using also okay with you passing URLs on their sites to your search provider? Do you ever accidentally paste confidential text in the wrong field?
Because there is no way to tell if the data is being logged (or, importantly, to prove that it isn't) it is prudent to assume it is, with an infinite retention— will your opinions about sharing todays data change if the search provider's behavior changes tomorrow? if some relevant government changes policy tomorrow? Does the fact that you can't ever take it back concern you?
These are some of the things I think of about it— and are why I wouldn't intentionally use such a feature regardless of how positive I feel about my search provider. And because of these thoughts— especially that I can't really make any binding commitment to future policy, or really guarantee that my policy is being followed (e.g. intelligence agency plants on staff) if I were a search provider I would not run such a service, unless I could figure out a way to build it such that I _couldn't_ turn evil, knowingly or unknowingly today or tomorrow.
It's exactly like using Fastmail. It's a convenience over self-hosted mail that sends all my email to Fastmail. Just like Omnibox, you have a choice to use it, and just like Omnibox, it sends a lot of personal information to a third party.
The reason people don't complain about hosted email is the same reason it's ridiculous to complain about Omnibox. Even though it gives information to the service provider, the service provider provides value on top of that information that makes a vast majority of people prefer it to the alternative, which people also have an option to use (by unchecking a check box or by running their own mail server).