Does "egoistic" and "egotistic" mean the same thing?
No. While similar, they have some important differences. Consider this note on usage from the Oxford Dictionaries:
The words egoism and egotism are frequently confused, as though interchangeable, but there are distinctions worth noting. Both words derive from Latin ego (‘I’), the first-person singular pronoun. Egotism, the more commonly used term , denotes an excessive sense of self-importance, too-frequent use of the word ‘I,’ and general arrogance and boastfulness. Egoism, a more subtle term , is perhaps best left to ethicists, for whom it denotes a view or theory of moral behavior in which self-interest is the root of moral conduct. An egoist, then , might devote considerable attention to introspection, but could be modest about it, whereas an egotist would have an exaggerated sense of the importance of his or her self-analysis, and would have to tell everyone
I've rarely heard "egoistic", but to me it indicates a shading of "egotistic" that suggests self-interest or being focused on one's self to the point of ignoring other persons, rather than of flaunting imagined superiority.
"Egotistic" and "Egotistical" are much more common and are typically used interchangeably.