I have always found this to be one of the most interesting arguments in the financial world.
The payoff of your mortgage as soon as possible, is, of course, the old way of looking at owning your home free and clear.
The pay it off method argument : "it is better to own the roof over your head than have the bank own it, and have mortgage payments", because if you fall on softer times at least you own your home and most likely your largest investment is secured.
The leverage point of view is the new way of looking at your home. This line of thinking is based on financing your home to use the funds to purchase other investments(usually real estate) to grow your money.
Both sides have excellent arguments for their case's and neither is right or wrong for everybody. Just like Automobiles, Homes, Mortgages, etc., one size does not fit all.
Client A does not like owning real estate(except for his home). He saves like clockwork, has a 401K and is on schedule(by making extra principal payments to pay his loan off 11 years early). He is already planning on going part-time(he likes his work), when he does so he can do more fishing and gardening.
Client B thrives on owning multiple properties and has increased the mortgage on her main home to obtain these dwellings. While doing so she has followed her own guidelines of keeping no more than a 80% percent loan to value on each property saving on pmi & reducing risk in case of a market downturn as well as keeping 6 months of reserves plus her regular savings. She plans on traveling when she is ready to retire and will use the income from those properties to help fund that lifestyle.
As you can see paying off the mortgage is in the best interest of client A and leveraging the equity in the home is in the best interest of client B.
Next time someone tells you what do with your home equity look at both view points, step back and then look at your finances, lifestyle, and life plan.
You will find the method that works for you.