The differences identified by the previous posters I think miss the really essential element that varies between academic and popular history-writing: the audience. Writing for different audiences then has several follow-on effects.

At their core, academics write for other academics; they do so to be part of a scholarly conversation. That means that academics read other academics, and they cite each other. Any academic work must be an original contribution, so you MUST read the literature out there or you risk replicating work. A well-organized paper will have a literature review of some sort, and sometimes a monograph will devote a whole chapter to this. Any academics who read your work (or hear it at a conference) will expect you to be well-versed in the broader literature and theory on a topic, and if you're not, or if you're doing something that's just totally out of left field, they're not going to be very receptive. They will also expect you to be familiar with and use certain data sets; if you submit an article to a journal, and you haven't at least done some archival work, or if you haven't consulted the "major" source base for a topic, you're not going to get much traction. And, if you're doing academic writing, your audience will expect you to make an argument about something, and it will expect to understand your reasoning and your evidence. Academic historians are, I think, not particularly interested in stories. We're not opposed to them necessarily, but a browse through all the major historical journals will reveal a LOT of articles that are argument-data-analysis, with perhaps an anecdote to tie it together. It will not turn up a lot of articles that are basically narratives.

The benefit of this for academic histories is that there's a formula to them, and if you understand that formula and the broader literature that something is engaging with, you can get a LOT out of an academic piece with a fairly small investment of time. You take an article or book, get a sense of where the author is coming from, what sources they're using, what theoretical positions they're taking, and then how it fits into the big picture. A good piece will have that information in the introduction, and if you want further details you can dive into the rest. I suppose a lot of people think that this is boring, but boredom is a matter of one's own interests. If I'm reading for work, as academic historians do, then I want the things I read to be as clear as possible a presentation of argument, evidence, and analysis. Narratives just get in the way, because the very nature of a narrative is that information is doled out in a certain sequence to illustrate conflict, build tension, and then resolve the conflict in some way. Now, there are some authors who are skilled enough to interweave narrative and argument (Bill Cronon, e.g.), but if I have the choice I'll take a clear argument over a narrative any day.

Popular historians, on the other hand, write for an audience of non-specialists. Their audience is not usually familiar with the broader literature on something, has little interest in theory, and no expertise in the data sets. As such, it's unnecessary for the popular historian to discuss these things, and thus these kinds of works typically stand on their own, and frequently make no or very little reference to other historians. And, because they're not fundamentally trying to advance a novel argument, data set, or interpretation, their writing doesn't have to be built around that structure. They are much freer to simply tell a story, because there's no bigger body of literature that they're trying to speak to, challenge, or build upon.

My field is History of Art but I can certainly point to one major difference between popular and academic accounts of history: There is a slight tendency to not let the facts get in the way of a good story. History is full of conflicting sources and one can spin a really entertaining story by being selective of the sources, or not quite fact-checking with contemporary academic researchers.

Just as an example, Andrew Graham Dixon is a famous art historian for the BBC, who is really primarily a presenter as opposed to a researcher. He wrote Caravaggio: A Life Sacred and Profane (2010) with a significant amount of glaring inaccuracies regarding Caravaggio's life-story. Almost everything about Caravaggio's life-story is contentious from his place of birth, to how far he traveled, to what started the infamous fight where he killed Tomassoni and had to flee Rome. Dixon had a narrative of Caravaggio's life presented completely as fact. Dixon certainly has the credentials but the aim of his book was largely to entertain.

I had a similar experience watching Oliver Stone's documentaries. It's hard to put your finger on it, but I reckon it comes down a couple of factors.

As /u/Steeznson puts it, it's basically about intent. The popular historian isn't so much concerned with providing an accurate and rigorous examination of what is 'true' as much as they are concerned with producing something readable/watchable for a mass audience.

What this is means is that they will be far more likely to present a narrative as though it were hard fact. This invariably involves the wholesale dismissal of other historiographies and facts. If they're even considered or raised at all.