Why use the fast Fourier transform for noise reduction instead of a classical electronic filter?
I'd like to know how to remove environmental noise from a speech recording.


Well it's  stored digitally now, right? so are you planning on putting your microphone next to the speaker after an analog filter to re-record it? 

Enough messing around, I'll be serious. 



In order to make a filter attenuate more in a smaller range of frequencies, aka making the frequency response curve more vertical, then you just need to increase the order of the filter. 

That is something that is reasonably easy to do in Matlab. It's also something that is feasibly to do post-processing. It's also about repeatability, if you apply the filter on a sunny day today, then you expect it to work identically to tomorrow when it's raining. You expect it to work exactly the same, right?

In analog circuits you have all these "5% resistor", "1% capacitor", and all other stuff. So if you want to make something exact you will definitely need to trim the circuit afterwards so it matches your desired filter perfectly. If you want to increase the order of the filter... then sadly.. it will make the filter so much larger physically. Instead of taking up the size of a credit card, it will take up the size of, I don't know, depends on filter order and what you're okay with. 

Regarding the repeatability, doing something today.. warm.. tomorrow.. colder... the resistances will change ever so slightly, the frequency response will change, a couple of Hz there, some there, the more components you got in your circuits, there more likely it is that your components will change their values. And then you have humidity, oxidizing... 

And here's the punchline that I should've said first, you can't really post-process it, unless you got cassette tapes. I'm not 100% sure what analog musical medium that is being used to record / delete easily. LP discs would be a nightmare... 

And let's not forget the price. One is software, if you write it yourself then it's essentially for free, the other requires components, physical parts. 

But don't think analog filters are bad, they got their uses, such as removing nasty harmonics in large DC motors, or making ultra silent stepper motors for 3D-printers by smoothing out the current. And tons of other uses. - Also if you would solve it with an analog filter, no one would think it would be a bad solution. 

I believe I'm indirectly answering why FFT is a better way to go about it, post-processing wise. The bottom line is that it's much cheaper to do. You could also just apply a notch filter if you know what frequency the noise is at. Or a wider, aka bandstop filter. 

And last thing I want to add... woaw this answer is so long, I'm sorry. But if you use an analog filter and you... mess up with your calculations and then think it's all fine and dandy and use it in some serious event, like interviewing the king of Sweden (Knugen). And you messed up with the sizing of a capacitor, instead of filtering 16kHz noise, you're filtering out 4kHz "noise". If you instead deal with it digitally then it's just a matter of changing some variables, you don't need to desolder -> solder another component. Also the interview is ruined.