The commonly accepted meaning of "constantly" in English is "without variations". So that "constantly changing" is a little bit of an oxymoron if you take it literally.
Keep in mind however that Ancient Greeks positively loved paradoxes as mind teasers and therefore sources of endless discussions (and Heraclitus is known for having added it's fair share to the corpus of Greek paradoxes).
Personally I would actually (paradoxically ;-) classify it as a tautology.
That's because the definition of "change" implies a measure of time, but at the same time, time is measured by change. So that stating that things change all the time is tantamount to stating that times passes by. Which it does, of course, "constantly" (as in "all the time", so to speak - if it stops now, you will never read this ;-).
However, Heraclitus quote must actually be replaced in its context. Living at the beginning of the Greek Golden Age, Heraclitus was supposedly reacting to mainstream descriptions of the universe he deemed too static.
To answer the second part of your question regarding the granularity of time: heartbeat-style or continuum like, this is still under discussion today and I don't believe anybody's got a definitive answer yet.
If we are to believe scientists, time can be modelled as a continuum (albeit distorted by gravity) in Relativist Physics but not in Quantum Mechanics.
Conversely the "heartbeat" kind of time seems to be more appropriate for Quantum Mechanics. My personal understanding is that this is due to the fact that Relativist Physics model certainties whereas Quantum Physics deal with probabilities.
But this is second-hand knowledge: as soon as lay people like me try to get a better understanding of these topics they hit the "equation wall". If someone can shed some light on this...
However, in the context of everyday life in English I guess you can take the continuous model for granted (... with a high probability ;-).