$projection vs $elemMatch

The difference in projection usage is somewhat subtle. In your example usage, these should be equivalent queries (in terms of index usage) but the $elemMatch example unnecessarily repeats the query criteria. The $ projection would be a more sensible choice for this example.

An essential difference noted in the documentation is the array field limitation for $ projections:

Since only one array field can appear in the query document, if the array contains documents, to specify criteria on multiple fields of these documents, use the $elemMatch operator.

Some further notes on the differences in the projection operators below ...

The positional ($) projection operator:

• limits the contents of an array field that is included in the query results to contain the first element that matches the query document.

• requires that the matching array field is included in the query criteria

• can only be used if a single array field appears in the query criteria

• can only be used once in a projection

The $elemMatch projection operator

• limits the contents of an array field that is included in the query results to contain only the first array element that matches the $elemMatch condition.

• does not require the matching array to be in the query criteria

• can be used to match multiple conditions for array elements that are embedded documents

The $elemMatch query operator

Note that there is also an $elemMatch query operator which performs similar matching, but in the query rather than the results projection. It's not uncommon to see this used in combination with a $ projection.

Borrowing an example from the docs where you might use both:

db.students.find(
    // use $elemMatch query operator to match multiple criteria in the grades array
    { grades: {
        $elemMatch: {
            mean:  { $gt: 70 },
            grade: { $gt: 90 }
        }
    }},

    // use $ projection to get the first matching item in the "grades" array
    { "grades.$": 1 }
)