I just finished a tutorial on the two methods of searching for tweets. Whenever this subject comes up on the Twitter developers mailing list, the usual response is that the streaming API is best, but that depends on your goals and programming ability. If you want to search for tweets in the past, or if you are not a very experienced programmer, the search API is the right choice. On the other hand, the streaming API will deliver tweets in real-time, which is very impressive for an app. I lay out all the pros and cons here.