General Thoughts
I have been using all kinds of wood to build the flutes. In the end I have not experienced a lot of difference in the tone. Even PVC flutes can sound very nice. A flute is an “aerophone” where the sound is primarily produced by the vibration of air inside the tube, not the material of the instrument itself. The shape and length of the flute’s air column primarily determine the pitch and sound characteristics, not the type of wood. Other instruments like guitars are resonating instruments, where the body of the guitar plays a crucial role in amplifying and shaping the sound. When the strings vibrate, they transfer those vibrations to the soundboard (the top of the guitar), which then resonates and amplifies the sound.
Here is a very interesting text on the subject: link
Another interesting “scientific” paper: link (pdf)
To summarize: There is a slight difference in sound depending on the material which might be important for you to find your “dream tone”. It is always a good idea to look at what seems to be the standard because generations of flute builders have probably used these woods for a reason. The most common are probably ceder, walnut, cherry and maple. Another good idea is to take a look at tonewoods that are used for building other instruments like recorders or guitars.
In the following sections I will talks about some aspects that might lead to a decision on which wood you would like to use.
Factors to Consider
Tone
Like I said, I believe the difference is there but not the most important factor in tone quality in my opinion. You should try different woods and find out which sounds best to you. Hardwoods have a slightly brighter, clear tone and softwoods tend to have a more mellow / warm tone.
Workability
There are several aspects that will influence how easy it will be to build the flute depending on your tools. Soft wood like cedar is really nice to work with even with hand-tools.
On the other hand the flute might be more brittle and get dents over time.
You can lookup the density and weight of different woods here (link).
Finish
Another thing to consider regarding the finish is how dense the surface is. If you have a lot of open/deep pores you will have to fill them before finishing or just live with the “natural” look and feel (also see workability).
Looks
Again a very personal decision. I like woods with a lot of texture but a bright, clear, shining wood can also be very nice to look at. A lot of people also laminate different woods to create some contrast.
My personal choices
I personally like a compromise of all the above which I found in Maple, black walnut and if I am brave (because of the red color) padouk. They look great, are hard enough for a durable flute and sound good to me.
Anyway … the choice of wood is a very personal decision and you should try different woods and get a feeling for them.
There is a great page online with lots of information regarding tone woods. Pleaser refer to the resource page (link).