They are different services which are intended to do different things.
I've been using D&D Beyond for a couple of years, as both player and DM. I've been using roll20 for a couple of weeks. I own many books offered through D&D Beyond, but not on roll20. So I can't offer direct feedback on every feature roll20 offers in this area, but the reasons I decided against buying content there were based on the paid features roll20 offers (as I understand them from how roll20 advertises them).
What are the trade-offs for purchasing an adventure module in Roll20 versus D&D Beyond?
There aren't necessarily tradeoffs, per se. Each offers what it offers, which may or may not suit your needs-- you'll never miss a feature of one service or the other that's not relevant to you.
D&D Beyond offers helpful features such as digital versions of all D&D books (which you can share with your players in various ways, with a relevant subscription), options for creating and organizing campaigns (and allowing or disallowing specific content within them), an encounter builder, step-by-step character builders, style-consistent templates for homebrewed monsters, characters, magic items, and more, all of which can be directly inserted into your games, and a variety of organizational features that help DMs and players get a handle on the large amount of information available in the books. D&D Beyond can also integrate playtest content, referred to as Unearthed Arcana, into your games. D&D Beyond has a comprehensive set of information about the game available, and offers tools for organizing that information in various ways.
Roll20 has some of the D&D books available, but not all of them. The content which is available can be deployed directly into games in various ways, including references (such as rules, stat blocks, artwork for character tokens or maps, and so on), drag-and-drop characters and monsters onto maps, and even create on-demand reference cards directly from the roll20 compendium. The content can be shared with players, allowing them to look up rules, spell descriptions, lore, and so on. Your ability to share information on roll20 is also dependent on subscription, similar to D&D Beyond.
Purchasing D&D content on roll20 offers a faster, easier way to deploy specific D&D content into a game than adding the information manually. A notable book which seems not to be available on roll20 is the Dungeon Master's Guide, which any DM will likely want access to in some form. This is in keeping with roll20's core purpose. Roll20 is primarily about running and playing games, and not so much designing them, planning them, or providing reference information for them.
Can content be shared / imported between the sites?
For owned content, like books, not as far as I know. You will not be able to buy a book on D&D Beyond and then use it in the roll20 compendium. There have been some community efforts to make some information portable through browser extensions and similar, but from what I understand this is limited to a few things, like character sheets. And I'm not sure I'd count community add-ons as part of either official service, but they do affect what value you can get from the service.
Depending on the content, you can manually transfer information from D&D Beyond to roll20. Without the roll20 versions of content, this can take a fair amount of work on the DM's part to set up.
Would there be a reason to purchase the module on both sites?
There are probably some edge cases, and it'll depend on your personal preferences. Maybe. The free artwork available on roll20 can be hit or miss. If you're playing a published D&D module, and if you're going to be buying content like art packs and maps to enhance your roll20 game, you may get more value for your money to buy a book on roll20 to get high quality, official content which is easily deployed.
The rub is that a lot of that nice, roll20-specific content is only available in bundles which include the roll20 versions of several official publications, and so your ability to mix and match products is limited compared with D&D Beyond. A lot of that content is nicely designed for roll20, so the extra expenses might be worthwhile for you.
I, personally, buy content on D&D Beyond because I find it to be a superior organizational and planning tool, and planning the games takes more of my time and effort than running them. The support is just more meaningful to me in that context than it is in running the virtual tabletop. I typically write my own campaigns; however, if I were to run a published module such as Descent into Avernus, I would be very likely to also buy that module on roll20 if it is available individually. The easy integration of specific campaign content, particularly maps and character tokens, is appealing to me.
Is there a common practice for using the two together?
I don't know if there's a "typical" way, but for my games my players use D&D Beyond to manage their character sheets and we use roll20 for tactical battles. It's basically the same as our in-person games, where we use D&D Beyond character sheets for the campaign and I manually draw on a battlemat or print out battle maps. This works well for us.
I can't speak to how well character sheets and other tools relevant to running the game (mechanically) work in roll20, but if all-in-one integration is important to you and your players it may be a valuable feature.