With a well engineered replacement window, you will not only save money but you will also be much more comfortable. Some of the higher end vinyl companies make some exciting products that look very nice, nothing like the low end stuff everyone is used to seeing a Home Depot. They also make composite windows that are made out of material similar to composite decking; Starmark is one.
In high end vinyl there is HiMark, Soft Lite, Okna, Sunrise, and Gorell.
I get this question all the time, there is really no substitute for a well engineered ,energy efficient window that's properly installed by a professional window contractor.

Good recommendations by mmarels.
Your question is a very valid one Cindy, and there is not always a right or wrong answer.
Can you significantly improve the efficiency of your existing windows through repair/refinishing, storms, etc? Certainly. Will it be as efficient as a quality replacement? No.
Now the hard part comes in when you start weighing the cost difference between the two (could be large, could be small), the condition of your existing windows, ease of use/functionality, the added value to your home by each option, etc, etc.
Generally speaking replacements offer great value when all is said and done, but there are certainly circumstances where a good storm window is a wiser investment, when your existing windows are still in good shape (operation and appearance). You may also look at other areas in the home such as insulation and air-sealing if "bang-for-the-buck" energy savings is your primary goal.

Here's the article that prompted me to post this question:

http://www.oldhouseguy.com/windows.php

Some of those points are legitimate, and some are very skewed, using questionable data. One main point is when he references the insulating capability of 3" of air between a storm and prime window. If that was truly sealed, dead air space that would be true, but its not. He also recommends heavy drapes/blinds over existing windows to save energy which again is true in theory, however in that case you are cutting off circulation of conditioned air to the interior window surface (making it colder) and increasing the likelihood of condensation on those old wood windows--if not on the storms behind them... There are plenty of other holes in there, but you get the point.
At the end of the day, as I mentioned earlier, replacement windows are the best option for some, but not for others. There are just too many variables to make a blanket statement without knowing all of the details in a given situation. :)

Even if my heating/cooling bills were cut in 1/2 with my new windows (100 year old house getting new Andersen windows next month) it would take 15-20 years for the windows to pay for themselves.

Saving the money will be nice, but don't spend $15,000 to save a few hundred dollars a year. Invest the $15,000 in a GIC at 3% and you'll earn $450 a year on it. Just an example.

I'm doing ours for comfort - our house is darn cold in the winter because of all the cold air streaming in. I have no illusions of the windows ever paying for themselves.

I think the deciding factor would be the condition of the prime windows and how much time or money it would take to rehab them and add that to the cost of a new storm. Sometimes it does make finacial sense to fix the old windows.Insulation uprades and air sealing are usually the better investment money wise. I would also look at doing inserts instead of full framed units.

Toronto Tim,

You're absolutely right. I'm probably considering doing it for comfort as well. Plus in the long run, it will increase the value of my home if I were to sell.

Exactly, seeing new energy efficient windows will make your home much more marketable.

Old windows can be weatherstripped. There are buyers who prefer the original windows. If you do replace don't throw away the old windows. You might also look at the Old House forum for more information.