If you forget the complex exponential part of the fourier transform (of which cosine is the real part) and just present it as

F(x) = ∫f(x)g(x)dx

for some arbitrary g, then the short time fourier transform is

F(x, t) = ∫f(x)w(x - t)g(x)dx.

So the mental model is your windowing the function you're taking the transform of. Remember that's just multiplication, so if w(x) is 0 outside of some interval, then so is f(x)w(x), and you use t to slide the window around.

But multiplication is associative, so you could easily think of it as windowing g(x) instead! Windowing the complex exponential gives you a wavelet. Rather than leaving it there, wavelet transforms add a scaling factor as well, giving

F(x, t, s) = ∫f(x)g((x - t)/s)dx

where g is now some 'mother wavelet'. Which could be the complex exponential, windowed or otherwise. If you remember that the complex exponential maps R onto the complex unit circle, with period 2pi, then increasing s increases the period (with respect to x), allowing for larger frequencies, and decreasing s gives you more detail in the frequencies you can see.

So, you'd want a wavelet that lowpasses the signal to avoid aliasing (which you can do by windowing!); and by the nyquist theorem in the discrete case you then need less samples to represent it fully. Taking that idea forward leads you to the whole filterbank thing you see all the time

I think the key thing is that wavelets aren't transforms of one variable, in the discrete case indices, but of three; index, time (location) and scale.