Often, there isn't a difference, but there are a few differences to keep in mind.
If your calculation references a field in another table, the calculation field cannot be stored. The auto-entered calculation can be stored BUT changes to the value from the other table's field do not update the value computed and stored in the field.
If you need a validation rule for your calculated value that cannot be set up to work on each individual field, an auto-entered calculation can be used and a validation rule set for it. A validation rule doesn't work for calculation fields. Your full name field is a classic example. If you need it to only store unique values (perhaps it will supply values to a value list), then it should be defined as an auto-enter calculation so you can set up a unique values calculation on it.