That's why I was a little confused myself. I'd read on MSE that it's better to overpay and reduce term rather than it reduce payments as you end up paying less overall if it reduces the term, at least that what it said on the MSE site.
Although the term of the mortage won't change, because that's the term you contractually agreed to, you will have ended up paying off the mortage before the end of the term if you never remortgage.
For example, my interest charge went from ~£85 to ~£75 a month when i paid off a lump sum. So now £10 a month more of more of my monthly payments are paying capital rather than interest plus I've also paid off £4.5k, so I will have paid the total I borrowed off well before the planned 25 years.
Otherwise as said above, the overpayments will help your choices on your next mortgage.
)
(Tin foil hat time!). For us, we did some quick sum and the rates are changing so much, we've spent so many years chasing good interest rates we're thoroughly bored of that and would rather not spend the mental capacity when we could just lump it into overpayments 
