the equals method tests equality right down to the millisecond.
I've just been doing some date/time calculations recently and remembered very quickly how much of a pain the standard java implementations are.
If you are ok using other API's then something like DateUtils from apache commons lang may be of use. It contains an "isSameDay" method.
Other than that you will probably need to use the java.util.Calendar to produce a Date with the non relevant fields (hour/minute/second/millisecond) all set to zero. Then you should be able to do an equals comparison on your 2 'day only' dates.