Chances are, when you first bought your house, you may have been asked by a lender whether you wanted to escrow your taxes and insurance or not. Usually to the new homebuyer, that begets a puzzled look of...huh? If you have been escrowing for a number of years, you undoubtedly know by now that your mortgage payment has gone up. Reason being, once per year, your mortgage servicer is required by federal law to update your account and make sure that they are not over collecting monies from you. Is there is a significant overage, they must return the overage to you. Generally this is not the case. What normally happens is that taxes go up, homeowners insurance usually goes up. You just forward the bills to your servicer, they pay them, and every once in a while your payment goes up 20 or 30 dollars. Your first reaction is WTF? You pay it and life goes on and one day you realize that your taxes have gone up 800 dollars and your homeowners 360 bucks. That is the typical experience of escrowing.
On the other hand, for those who don't, or do not wish to escrow, a different situation often plays out. Once you bought your house, you paid the original tax bill and bought a homeowners policy with a check. Hopefully you set out with some type of plan to budget 1/12 of those bills in a special account so that when next years bills come do, you can just withdrawal the money, pay the bill and all is well. What sometimes happens is the "out of sight, out of mind" theory, followed by the "didn't I just pay that". Yes, you did...last year. Unless you have iron discipline, it can be challenging to try and save the money each year. Even if you are successful for a while, things come up. It's there to dip into for car tires, home maintenance, etc all with the good intention of replacing it as soon as you can because taxes arent do for a while yet.
My final word on this one, from my own experiences as both a homeowner who has done both and as a lender is, "When in doubt, go the escrow route."