Two different meanings of "myself".
εγώ ο ίδιος = I myself; I and nobody else; I personally.
εαυτό is perhaps best translated as "self" -- it's always (as far as I know) used together with personal pronouns and is used when it's not the subject (most commonly the direct object, but could also be in a prepositional phrase) to refer to the same person as the subject when the subject acts on themself, like with a reflexive verb.
You can even use both together: Εγώ ο ίδιος αγγίζω τον εαυτό μου "I myself touch myself", i.e. "I (and nobody else) touch myself (i.e. subject acts on the subject)", Εσύ η ίδια αγγίζεις τον εαυτό σου "You yourself touch yourself" etc.
Note also that ο ίδιος inflects for gender and number (and case), but εαυτός only for number and case - Εμείς οι ίδιες αγγίζουμε τους εαυτούς μας (not: τις εαυτές μας), for example, when said by a group of women.
There's also a third meaning of "myself" which is μόνος μου (etc.): I did it by myself, Το έκανα μόνος μου, Can you do it by yourself? Μπορείς να το κάνεις μόνη σου;