In short:

Box is a more powerful, convenient, and potential replacement for div. You can change it to any HTML elements (e.g. span), but most of the time you will use it as div replacement.

Grid is the syntactic sugar of Grid Layout.

Use Box when you want to group several items and control how they look on the page. For example, you can take several paragraphs and use a box to put a border around them.

Use Grid for setting up a grid layout system for organizing content in columns on the page. Designers will divide the page into 12 columns with the idea that it is more visually appealing to have content aligned along each column or group of columns. Here is an article that provides much better detail on the subject: https://www.interaction-design.org/literature/article/the-grid-system-building-a-solid-design-layout

Box

The Box component serves as a wrapper component for most of the CSS utility needs. The Box component packages all the style functions that are exposed in @material-ui/system . It's created using the styled() function of @material-ui/core/styles

Grid

GridBox is the low-level representation of Grid. Except for low-level notebook expression manipulation, GridBox should not need to be used directly. In a notebook, columns of a GridBox can be added using and rows using . or a menu item can be used to start building a GridBox.

Box

The Box component serves as a wrapper component for most of the CSS utility needs.

The Box component wraps your component. It creates a new DOM element, a <div> by default that can be changed with the component property.

Let's say you want to use a <span> instead:

<Box component="span" m={1}>
  <Button />
</Box>

This works great when the changes can be isolated to a new DOM element. For instance, you can change the margin this way.

Grid

The Material Design responsive layout grid adapts to screen size and orientation, ensuring consistency across layouts.

The grid creates visual consistency between layouts while allowing flexibility across a wide variety of designs. Material Design’s responsive UI is based on a 12-column grid layout.

How it works

The grid system is implemented with the Grid component:

• It uses CSS’s Flexible Box module for high flexibility.
• There are two types of layout: containers and items.
• Item widths are set in percentages, so they’re always fluid and sized relative to their parent element.
• Items have padding to create the spacing between individual items.
• There are five grid breakpoints: xs, sm, md, lg, and xl.

Further Reading

Docs on Box & Grid

The grid creates visual consistency between layouts while allowing flexibility across a wide variety of designs. Material Design’s responsive UI is based on a 12-column grid layout.

How it works

The grid system is implemented with the Grid component:

It uses CSS’s Flexible Box module for high flexibility. There are two types of layout: containers and items. Item widths are set in percentages, so they’re always fluid and sized relative to their parent element. Items have padding to create the spacing between individual items. There are five grid breakpoints: xs, sm, md, lg, and xl.

A Box is just that, a box. It's an element wrapped around its content which by itself contains no styling rules nor has any default effects on the visual output. But it's a place to put styling rules as needed. It doesn't offer any real functionality, just a placeholder for controlling the styles in the hierarchical markup structure.

I often think of it as semantically similar to the JSX empty element:

<>
    Some elements here
</>

But just with some Material UI capabilities:

 <Box component="span" m={1}>
     <Button />
 </Box>