Well, there is simply no best time to visit Paris, because each season has its advantages and its drawbacks. It all depends on your motivations and priorities.
As much as it's enjoyable to visit Paris in the summertime when almost all the Parisians are away and Paris turns into a big amusement park where almost all the streets are filled with tourists looking at big maps of the city trying to figure out how to get from one main attraction to the other, that's also the time of the year where it can often be quite hot and the museums are overcrowded (except the ones that are off the beaten track, of course).
Curiously, one of the busiest times of the year is around Christmas time and New Year's eve. One of my worse experiences in 33 years of guiding (time flies) is having had to line-up for 30 minutes with my group on New Year's Eve just to access the room of Mona Lisa. Unbelievable!
For me, two of the best months to visit Paris are:
September (the first two weeks are the best): because it’s back-to-school in most European countries and things are kind of slow while the weather is supposedly still nice.
May: because you’ve heard of Springtime in Paris and also because it’s a bit quieter compared to the big masses of Tourists during June, July and August. Make sure to check the dates for Easter first: you don't want to visit the Paris museums on your own one week before or after the Easter week-end!
July is not a very good time to visit Paris, for the very simple reason that it’s the month I normally go away on vacation! July is indeed the time I enjoy life in my flat on the Riviera because it's the best time to go there. But then again, if you can only come to Paris in July (I know a lot of you travel in July), I can probably recommend you to one of my colleagues or, if you intend to hire me for a few days and/or if you intend to also visit the French Riviera and would like to hire me for that as well, I guess we can work something out and I can maybe move my vacation to other dates...
SALES: good to know: in London, the sales start on Boxing Day but in France, by decree, the winter sales start the first Wednesday of January and they last for 6 weeks so, if you want to do some shopping while you’re here, that would be a good time to come. And, while we’re at it, summer sales take place mostly the whole month of July and the first week of August.