Which is the least expensive way to pay off a home with  interest rates and
number of years to pay being the same, a regular fixed rate mortgage or a
home equity loan?

Given that the interest rates are equal and the number of years are
the same, the 2 loans should have the same payoff costs.

The terms of 1 loan may be different from the other:
1) One loan may require mortgage insurance.
2) There may be fees or points associated with acquring a mortgage.
3) The home equity loan may be variable (probably not from your
wording, but potentially).
4) There may be a penalty associated with paying off one of the loans quickly.