I would say this depends on the department and their hiring practices, but in my department (math at a public US research university) applying early can definitely give you a leg up. In math in the US, there is a centralized application system (mathjobs.org) that most places use, and in my department all faculty get access to start looking at applications well before the deadline. Many of us do start looking at applications early, partly out of interest, and partly so we can take our time.
The advantage of applying early is that faculty may start talking about some candidates before any search committee meetings and people have the chance to get excited about your file early, which would give you some early momentum to help you stand out when we officially have meetings and rank candidates.
That being said, in other departments, faculty might not look at applications at all before the deadline, in which case there is no real affect. I do not know how common our approach is, but at least a few of my colleagues agree with me that it is better to apply early. I have also heard of departments that have somewhat misleading deadlines (they start officially reviewing files before the deadline, which might be in fine-print in the ad), so my advice is to try to prepare you application a bit early, and then submit application when your portion (i.e., excluding the references) is ready.
Incidentally, in mathjobs, the candidates submission date is readily visible, though it is not something that we focus on or analyze. (Though often we'll sort by submission date to see what files came in since last we looked.)