I agree with november _moon; it depends which is most important to you. I book places to stay first, as Where I stay is a priority for me.
If your dates are absolutely inflexible, or of you are using miles, booking air first makes sense.

Also important is what is relevant for this trip on top of what is relevant to the person. For those living in cities where flights to Paris is plentiful, the flight may be a subordinate priority. But that is not the case for everyone. It would be best to weigh pros and cons of different approaches for each trip. Even going to the same place at the same time of the year, the answer may be different from year to year. If the value of a trip hinges on the particular property, then the apartment is the priority. If you can live with an alternate accommodation, then the priority is different.

Great points! Airfares are so unpredictable these days...I'm wishing for an even better fare to come along soon.

As far as my apartment, I always had the idea of one with a view overlooking the eiffel tower. HAD to have it. Little money. So I searched for over a year and found the perfect one (sans the no tub just a shower issue lol) within financial reach. Im sure when I go back I will go even cheaper and not have such strict demands. I guess it is as everyone said, what is most important to you?

Thanks everyone for your insight. I decided to go with BeachGirl247's advice - to book the flight and get the best departure/arrival times. I want to use FF miles on a direct flight - so I will book flights first before apartment. But apartment will be a close second!