They are synonyms. They both mean "to start/to begin an action " . Though there are some cases where we would use only one of them. A scenario for example: You have an appointment with a friend of yours and you're a bit late, he calls you to see if you're on your way but you just got in the car, then you will say (to show that you just got in the car and you're about to go) : " Τώρα ξεκινάω (για να έρθω)" , you wouldn't say "Τώρα αρχίζω". Though in most cases we use either one of them. I hope that helped! :) Have a good day.

Περίπου ναι. Κανονικά "ξεκινώ". Αν δώσεις παράδειγμα, θα σε κατατοπίσω καλύτερα.