I have served on many cycles of graduate admissions, and I have NEVER considered the submission date as indicative of a student's interest in our program (even late applications are usually given the benefit of the doubt). The exception to this rule may be programs that receive hundreds of applications. If committee members try to get ahead of the process by reading the early submissions early, then submitting early may help. However, a complete application includes both your materials and the letters of recommendation, the latter of which often arrive a few days after the deadline. So, my recommendation would be to continue fine-tuning your application so that you have the best representation of yourself in your submitted materials. Of course, do keep track of the deadlines so that you don't submit late!