This is a sticky question. Preferences for academic titles varies between countries, institutions and individuals.
In the US, any individual employed in an instructional capacity by an institution of higher education can be properly addressed as "Professor Smith", even if his or her official title is "Assistant Professor", "Lecturer", "Adjunct Instructor", etc. Moreover, any individual holding a doctoral degree can be addressed as "Dr. Smith".
Of course, these two cases often overlap, and you are left to decide whether to use "Professor" or "Dr.". The pattern I've observed is that at institutions where only some of the faculty have doctorates, those who do are more likely to prefer "Dr."; whereas at places where almost everyone has a doctorate, they will all prefer "Professor".
I think the safest default, and the one I most commonly see, is
Dear Professor Smith,
I would not abbreviate "Professor" as "Prof.". It sounds too much like an annoying student saying "Hiya, prof!"