There have been many articles related to just how much new windows will save on energy costs and NONE have concluded they are big savers. Payback is usually out in the 20 year range and with today's lower energy costs it may be longer now.
Having said that I have replaced most (eventually all) of my old double pane double hung windows as part of my insulation upgrade including new siding. The timing was right and the wife was tired of having ugly hard to open windows. I'll never save enough from just the windows, but they are an excellent upgrade given the effort involved with all of the other improvements.
Sometimes you just want new and every time I open one of our new windows with one finger I'm happy.
If you have old single pane windows you will benefit more. But that article is misleading. If you add up all of the air leaks in a typical home there is a 12" x 12" in the side of your home. Of that, windows contribute very little and, and, you need all of those leaks to maintain air quality. Recommended minimum air exchange is 1/3 of all inside air every hour. Tighter than that and you develop moisture issues and low quality air.
The explanation gets longer but consider your window upgrade for reasons other than saving energy. I like mine and I'm an energy auditor (retired).