Years ago I lived in Hong Kong and traveled to Beijing often. Both cities have changed a lot over the years, so their appearance will be different from when I was there, but some things will remain the same.
First, the weather. September can be a very nice month in Beijing. It's typically a little cooler and drier than the summer months. However, the air quality is poor, so your photos of the Temple of Heaven and Tianamin Square probably won't show beautiful blue skies. There are lots of places to visit in Beijing proper, and you can make a day trip to the Great Wall.
In HK it will still be warm and a bit humid, but not miserable. If you want to spoil your wife you can stay at one of the high end hotels. I really liked the Conrad on the island side. There is a ton of stuff to do in HK, and you could easily spend a week there and still have lots left to do at the end of your visit. You can get around on the MTR (fast and efficient, also cheap). You'll probably want to avoid taxis because traffic can be very slow. Some of the things I enjoyed: Kowloon night markets, dim sum (ask your concierge for a recommendation), bird cafes in Kowloon, jewelry shopping, fabric shopping (I know that sounds crazy, but we found fabric "alleys" that sold the most beautiful fabrics, and had many blouses, skirts, and dresses made for my wife), a day trip to Guanzhou to see the Quinping Market (amazing market where you can buy every manner of bug, fish, amphibian, mammal, and plant, all for the dinner table, exploring the old temples is cool, walking the old neighborhoods, taking the high speed ferry to Macau for a bit of gambling. I could go on and on.
The guide books are full of good info. Buy a couple, read the descriptions of both towns, and make your decision based on what sounds good to you. Have a great trip!