Difficulty in theoretical understanding of Newton's law of cooling

but what I do not understand is why is it specified that there is a breeze?

Without seeing the statement in context, it looks like they are only talking about forced convection. But Newton's Law of Cooling also applies when there is no breeze, which is natural convection due to the natural movement of air when it is heated or cooled.

That aside, the main difference between the constant of proportionality in Newton's Law of Cooling and Fourier's Law of Conduction, is that $h$, the convection heat transfer coefficient of the former is a variable whereas $k$, the material thermal conductivity of the latter is a property of the material, a constant.

$h$ depends on the movement of the fluid (e.g, natural convection vs forced convection) and the physical characteristics of surface (e.g., rough vs smooth).

Hope this helps.