The style and sx prop in MUI components pretty much do the same thing. The sx prop offers a few shorthand syntaxes, and allows you to access the theme object. But apart from that, they seem identical. When are you supposed to use one over the other?
To really understand which one to use, we need to understand what's happening under the hood. Material UI uses emotion(, or whatever styling engine you chose manually), in order to style its components. On the surface, the following two might seem to be doing the same thing:
<Box sx={{ height:'50px', width:'25px' }}/>
<div style={{ height:'50px', width:'25px' }}/>
Both render divs with the required height and width, to the DOM. But for the div, the styles are applied as inline styles, whereas the Box applies the styles in the form of a class, to the div element. The class definition itself, is stored in the head tag, which you can inspect, to see the following

This is all well and fine, as long as we're declaring the styles only once. But stuff really goes crazy when you add dynamic styling. Perhaps there is a state variable controlling the height of your div.
function CustomComponent(){
const [computedHeight,setComputedHeight]=useState();
useEffect(()=>{
window.addEventListener('resize',()=>setComputedHeight(window.innerWidth/2))
},[]);
return (
<Box sx={{ height:computedHeight, width:'25px'}}/>
<div style={{ height:computedHeight, width:'25px'}}/>
)
This is a common scenario, where some external variable(the width of the browser window for eg.) determines some property of your component. What happens is every time this state changes, MUI creates a new class, sets it as the class for the Box, and adds the definition to the <head> tag of your website, as a brand new <style> tag. So in very little time, the head tag fills up with hundreds of style tags, which is clearly undesirable.
However, in the case of your div tag, the styles are located inline. So no matter if the value changes once, or a hundred times, there is only one definition of the style, and it exists on the element itself. No need to worry about it.
TLDR; Put your dynamic styles(the ones that change based on some variable) in the style prop, and put all the static styles in the sx prop