Another thing I do is send my taxes in a month early, in case I make a mistake, they will let me know instead of fining me. If you're getting a refund that doesn't matter though (because you won't be fined for being late if you're getting a refund, April 15th is not actually the deadline for a refund).
Actually it can make a difference, but you are correct, it doesn't affect how much money you're due (if you have a legitimate refund due you).
IANACPA, but my wife is (and prepares personal tax "during the season").
In some states (and the Fed), the pool of money available for refunds often gets depleted mid-way through the tax season. Submitting early gets your refund processed before the state/fed need to replenish these funds, which can sometimes get you your money faster.