The first is used when you want to refer to an entry in a list or Matrix.  This is referred to as an index subscript.

If a list is defined by

Then you can select the second entry using square brackets.

Alternatively, you can use subscript notation

which is found on the Expression palette.  Note the tooltip for it shows the equivalent square bracket notation.

If a matrix is defined by

We can select any entry using M[i,j]

or using the subscript notation $M_{i\,j}$.

The shortcut key to enter an index subscript is Ctrl + Shift + _ (Command + Shift + _, Mac)

For more information, see indexed.

You can change the shortcut key for index subscripts to a simple underscore, if desired.  For details, see Underscore Entry in the Options>Interface help page.

The second type of subscript is a subscripted variable name.  In the example below, $\mathrm{x\_\_1}$ is a name that is not related to the name $x$.  This is sometimes referred to as a literal subscript.  To enter a subscripted variable name in Maple, use the second subscripted name shown in the Expression palette, .  It is shown all in one color.

To enter a subscripted variable name using a shortcut key, type two underscores (__).

Note we have not changed the entries of the list $x$ because $\mathrm{x\_\_1}$ is simply a subscripted variable name that is not related to $x$.

In Maple, subscripted variable names are one kind of atomic variable.  There is an option in the View menu that displays atomic variables in a different color, to visibly distinguish them from other math.  For more information, see atomic variables.

Here, we form an indexed name, $y_i$ and then verify it is of type indexed.

Next form a subscripted name using double underscores.  

 The subscripted name is of type symbol.

For more information, see type/indexed and type/symbol.