Basically, if you want to search your calendar on the iPhone, you'd better be using the native calendar app. Why? Because other calendars you can run on the phone -- such as Google Calendar -- incredibly don't provide a way to search the calendar! I couldn't believe this when I first discovered it. After all, it's trivial to search your Google Calendar on a regular Web browser via the prominent "search my calendars" button at the top. But Google Calendar as it runs on the iPhone has no such feature.
Now in fairness, Apple only added this ability to the native iPhone calendar with the recent release of the iPhone 3.0 software that came with my iPhone 3 GS. So it's not like the iPhone could do this at all before that. But given the immense popularity of the iPhone, it's critical that Google add this feature to the iPhone implementation of its own calendar. After all, Google is a search company!
We'll probably see Google and other cloud calendar providers fix their iPhone implementations before it becomes easy to sync the native iPhone calendar directly to their cloud calendars, for reasons explained previously here.