Hyatt is generally better than Hilton and Hyatt diamond is better than Hilton diamond. But it really is more about the individual hotels in a circumstance like this.
If I were you, I'd look at the flyer talk reviews of the specific hotels I'm considering or ask on here. In the FT reviews people usually comment on their treatment as a diamond, so you can get your bearings from that.
We had a wonderful stay at the Grand Hyatt Tokyo. Probably best we've ever had. The little touches and consideration were just top notch. However, I found that customer service in Tokyo in general was very thoughtful, so my experience doesn't really help you decide, since the service and experience at the Hilton might be great too.
Seems like the obvious difference would be if you have more than two people in the room, your free Hilton breakfast probably won't cover everyone (unless someone else in the room is also an elite, as helpfully pointed out above - I didn't realize that)
(Fixed some typos)