I believe some mortgage providers only credit additional mortgage payments on an annual basis, in which case, you could be paying down additional sums on the mortgage for a whole year before you get any benefit for it. Others will allow you to overpay on a monthly basis and every month's payment to go towards reducing your eventual term and the amount you owe. As Hymie says, you need to check the terms. Did you not have any paperwork documenting this when you first took out the mortgage?
Given the current low rate of interest on savings and the fact that we're in an uncertain economy, it probably makes sense to use surplus monthly funds to reduce your mortgage. However, it's also recommended that you keep between 3 and 6 months living expenses in a savings account in case of illness, redundancy etc.
There's no doubt these days that as you grow older, job security often diminishes for all kinds of reasons - changes in techology, recessions, company takeovers, etc. Knowing that you could be mortgage free at a much earlier point in time could be a great relief in allowing you to sleep more peacefully at night if your career hits a low patch later in mid-life and you find yourself unemployed or having to take a lower paid job to survive.
Obviously the earlier in the life of the mortgage you start making extra payments, the more impact you will have on reducing your debt. Clearing down and overpaying as much as possible while interest rates are low will cost you less in the long run than trying to overpay when interest rates are higher because then more of your overpayments will actually be directed at reducing the interest owed rather than the capital.