I've scoured the internet trying to understand how various characteristics of instruments (Bell shape and diameter, bore size, material, finish, construction) in the trumpet family (Trumpet, Cornet and Flugelhorn primarily) effect their tone and playability (Range, Clarity of Articulation). I've learned a lot but I still can't figure out what physical characteristics of a Flugelhorn cause it to sound so different from a Cornet. My understanding is that a Cornet is "Darker" sounding than a Trumpet largely due to a greater percentage of conical sections of pipe as compared to a Trumpet, but don't flugelhorns also have this conical characteristic? True a trumpet has more bright overtones than either of them, but the way that each instrument differs from a trumpet is slightly different to my ear. People often describe a Cornet as being "Dark" and a Flugelhorn as being "Warm" but I'm not sure if they mean to use the terms interchangeably or if there is a tangible difference to the sound. To my ear the Cornet has a more focused tone, while the Flugelhorn's sound seems to sort of feather out? So to summarize, how would you describe the differences in the sound of a Flugelhorn and a Cornet and what part of the instrument is causing the difference, is it the larger bell? Slow vs Fast taper? Bore Size?
Flugel is far more conical than cornet. Cornet is only just barely not cylindrical and is really just slightly sweeter than trumpet, while flugel is so dramatically conical that the design involves moving the valve cluster way up so that the bell can take up the majority of the instrument, and is more akin to a euphonium. That's what's responsible for the tone difference.
Another key reason for tonal variation between the two is the mouthpiece design. Although a typical true cornet mouthpiece is generally deeper and more funnel shaped than a modern trumpet mouthpiece, it is less so than a typical flugel piece. Other factors include bell size, the way the bell flares earlier and faster on a flugel than a cornet, the way the leadpipe generally goes directly into the valve block only a few inches behind the mouthpiece, and a lot of other factors, including Matt Putnam's earlier answer.
Similarly, if you put a 3C mouthpiece with a cornet shank in a Cornet, it will sound much more like a trumpet. If you put a Wick 2 in it, it'll sound like a traditional brass band cornet sound.
Flugel mouthpieces are often (but not always) much deeper than a cornet mouthpiece and with a larger throat as well.
Yet another difference is the bore size of the lead pipe. It's not intuitive, but thinner lead pipes give a softer sound.
Flugelhorns have different standards for size of lead pipes. The ones you find in the US often have larger lead pipes than the french ones, which have a softer sound.
Consider the french horn, it has a much softer sound than a trumpet, the lead pipe is much smaller.
I could not fit my trumpet mouthpieces in my flugelhorn, they are too large.