• Guaranteed Returns:  If you have a fixed-interest mortgage, you can always count on the return of your prepayment investment to remain constant.  If you have the extra funds to spare, but think the market is too risky, it may be worthwhile to apply that money to your mortgage.
• Cancel Your Private Mortgage Insurance (PMI):  If you had to borrow more than 80% of your home's appraised value for your mortgage, then you're probably paying private mortgage insurance, an extra fee that could cost you as much as $100 a month.  By making extra payments, you can obtain the required 20% equity in your home and convince your lender to drop the PMI.
• Peace of Mind:  Paying off a mortgage is a long-term financial goal for some homeowners, and for them, that peace of mind carries an enormous emotional value.  Just make sure that your high-interest debt and retirement savings are taken care of, too.
• Retirement Beckons:  Paying off your mortgage early can mean a big reduction in living expenses after you're retired.  And because you won't be forced to tap into your retirement savings in order to make monthly housing payments, you might be able to live a little larger, or retire a little earlier.

On the first point: there are investments that are not that risky which yield decent returns. More money paid towards the mortgage does not yield any return and a small one is better than none. (I know some will say it "builds equity" but technically equity is useless unless it is used and it can be used in constructive as well as destructive ways.)

On the second point: yes the PMI can be cancelled or it could be paid upfront and thus be tax deductable unlike when paid monthly. 

On the third point: if it is possible to take extra money and put it into investments that compound, then the money can work for the person where paying extra payments will not.

On the fourth point: one of the reasons I noted what I did in the other three points above is to help facilitate to some extent the goal of paying off the mortgage earlier. One can either work for money or have money work for them. And the greater the extent of the latter, the more security one will have long term because money can work when you cannot. 

Now to touch on the listed advantages...

 How Not Prepaying Could Be Beneficial: 

• Market Return:  If you can get a better return, and you're willing to take a long-term ride on the market, it may be more profitable to invest in an index fund.
• Interest Tax Deduction:  While making extra payments reduces principal on your mortgage, it also reduces the amount of interest you can claim on as a tax deduction.  If you're in a high tax bracket, being able to use this deduction to lower your taxes might be motivation to invest your extra cash elsewhere.
• Prepayment Penalties:  Before increasing your monthly payment amount, make sure your mortgage does not have any prepayment penalties.  While most fixed rate mortgages don't carry such penalties, they may be attached to adjustable rate mortgages.  Talk to your lender for details.

I agree with point one. Frankly, people should make themselves willing to invest and learn about it because otherwise they will never get out of the rat race. On the second point, yes it reduces the amount used as a tax deduction and that is why it is rarely a good idea to do. (There are exceptions of course but as a rule the interest deduction is either the best or the only one most homeowners have.) On the third point, well said. Prepayment penalties can affect the matrix of the equation also -the advantages and disadvantages of this vary by the program and the circumstances of the borrower.

Good lists Tony!!!