Some of the biggest differences are:
(1) Exponent is a free open source project, and we don't try to charge developers to use it. (Titanium is open source now but they try to make money by charging people for premium features--which is totally fine, but just different from what we do)
(2) Exponent is based on React Native which is newer and people seem to like better (search Google for blog posts about experiences using React Native and then search Google for blog posts about using Titanium), though I can't actually compare them personally since I don't know that much about the inner workings of Titanium.
(3) You can do instant app updates with Exponent since the JS is run on the device rather than being cross compiled to native code. I don't think you can do this with Titanium (though I may be wrong -- they've added a lot of features since I last checked on it)