I guess they mostly can be used interchangeably, but they are not exactly the same.
Einschätzen contains the word estimate, so it doesn't sound as definitely as a Beurteilung. After someone had a test and doesn't know his result yet, it would be more natural to ask:
Wie schätzt Du Dein Ergebnis ein?
than "Wie beurteilst Du Dein Ergebnis?" - because you normally would want to know the results before you beurteilen it. On the other hand, if you already know the results and someone asks "Wie schätzt Du Dein Ergebnis ein?" - he probably wants to know your thoughts about how well you did compared to others. So Einschätzen could also imply a comparison or maybe a ranking.
If you participate in a race, you could get asked before it starts:
Wie schätzt Du Dich ein?
meaning what you think how you will do compared to the others. You would not get asked "Wie beurteilst Du Dich?" It would have to be something like
Wie beurteilst Du Deine Chancen?
Generally, I think beurteilen is not often used for persons. You would einschätzen people, but normally not beurteilen. You would beurteilen their efforts, their results, their achievements, but probably not themselves as persons. But I guess nobody would cringe if you would use it.
Last but not least, there's 1 meaning of einschätzen which can never be replaced with beurteilen and this is in context of taxes. The tax office can make a Einschätzung for you and this is a fixed term.