The answer, like almost every college admission question, often depends on the school. At a selective institution, there is typically no preference given to applications that arrive early, as long as your application is postmarked by the DEADLINE.

Some schools have “rolling admissions” which means that applications are reviewed and decisions are made as the applications come in. In this case, it is valuable to send in your application as soon as you finish - it shows your enthusiasm for the school, as well demonstrating that you “have your act together”. Additionally, there is the added benefit of knowing your decision early (many rolling admissions programs will give you a decision within a specified number of week after receiving the application).

I would NOT recommend waiting until the last minute. You'd be surprised at how many thousands of applications arrive the week of the deadline (most postmarked ON the deadline, or the day before). It takes time to process an application. Steps can include opening the applications (or downloading them), data entry, processing, score checking, filing, and a variety of other time-intensive tasks. Let's say the deadline is January 15. Send your application in December and it is likely to get processed fairly quickly. Some schools have systems where you can check your application status to see if your scores have arrived, if your recommendations have been received, etc. But… send your application on January 15 - and your application is likely to get backlogged with the thousands of others that arrived on d-day. While this won't effect your decision - it can make it more difficult to check on the completion status of your application and ensure that all of your materials have arrived and been added to your file.