The names of these classes of instruments suggest that the material from which they are made is significant but this is not so. Flutes are frequently made of metal and saxophones always are yet they are classified as woodwind.
(Note that I play the clarinet and saxophone but no brass instrument so my knowledge of those will be weaker.)
So, what is the defining characteristic?
The form of the mouthpiece. As far as I know, all brass instruments have similar mouthpieces. Woodwind instruments have at least three distinct forms: flute / piccolo, single reed (e.g. clarinet and saxophone), double reed (e.g. oboe and bassoon).
How notes between harmonics are obtained. Brass instruments adjust the length of the tube with valves or slides. Woodwind instruments open or close holes in the tube with fingers or keys.
Whether the fundamental note is used: rarely on brass instruments but routinely with woodwind.
As far as I know, these properties go together. For example, I don't know of any commonly used instrument with a brass style mouthpiece but woodwind style holes in the tube for pitch control. It has been tried long ago but it was not successful: Keyed trumpet (Wikipedia).
The use of the fundamental could be significant. If it is not used then the valves just to fill in a gap of a fifth between the second and third harmonic. If the fundamental were used then an octave or more (clarinet) would be required and this might be hard to achieve with valves.
It might be hard to add valves to a wooden instruments though, of course, not all woodwind instruments are made of wood.
It has always been a puzzle to me that the holes in the tube technique produces a poor effect with brass instruments but satisfactory with woodwind instruments.
Edit: I forgot the recorder in point 1.