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I would say that the difference is: Ελένη is the word "Eleni" while η Ελένη is the person Eleni.
So you might say Το όνομά μου είναι Ελένη (my name is "Eleni") but Εγώ είμαι η Ελένη (I am Eleni).
Proper names are nearly always used together with the definite article when referring to a person.
How do I know when to use an accent.
In Greek, you always put an accent mark on the stressed syllable on all words of two syllables or more.
One-syllable words do not have an accent mark, with a handful of exceptions, where the accented and unaccented versions have different meanings. (In this, it's similar to French ou versus où or a versus à, where the accent mark has no influence on pronunciation and only serves to distinguish two otherwise identical words.)