A few years ago I drove a friend’s car. The accelerator was much more sensitive than I was used to, and it caused a jerky ride: every time I touched the gas, the car lurched forward. It was a different experience driving a moving van full of heavy furniture. No matter how much I leaned into the accelerator, the speedometer crept upward with painful slowness.
The car I’m used to driving is somewhere in between—it’s acceleration isn’t quite as zippy as my friend’s car’s, and not as sluggish as the truck’s. With woodwind instruments it’s important to have a similar balance.
Response is how readily the system (you + the equipment) produces tone. A very responsive setup/technique will make a sound with the faintest whisper of air.
Stability is, in a way, the opposite of response. Rather than responding to the slightest puff of air, a more stable setup has some “cushion”—you lean into it a little more to produce a tone.
A very responsive setup takes less physical effort to make a sound, but the sound can be harder to control. The pitch and tone are more flexible, which can be a good or bad thing depending on your preference and playing situation.
A more stable setup takes more effort to produce tone, but it tends to have more steady pitch and tone. Again, this is a tradeoff.
For most players and situations, some kind of middle ground is the right choice: enough response to articulate notes at pianissimo, but enough stability that you don’t have to devote all your attention to keeping things in tune.
Assuming your tone production is a well-oiled machine (breath support, voicing, and embouchure all working well), your equipment choices and condition play an important role. That means matching reed/mouthpiece/headjoint to your instrument, and keeping pads and tenons in good non-leaking condition. For example, a saxophone that blows very freely (or, in other words, is very responsive) may need a little resistance in the reed and mouthpiece (to provide stability). A flute that has a lot of resistance built in may need a freer-blowing headjoint (for ready response).
Develop your basic tone-production technique and make smart, reasonable equipment choices to find the response and stability you require.