Prequels pose a rather unique challenge to writers and filmmakers. Not only do they have to tell their own coherent stories, but they also have to adequately set up the events of the installments that came before them. Since they take place earlier in the series, they often can't include a lot of familiar elements from their predecessors, yet they still have to be similar enough to those predecessors to keep from alienating fans. What's more, most books, films, and so on are created without any prequels in mind, and this lack of accommodation can really plague a prequel's storyline and form plotholes and inconsistencies in the overall series.
Peter Jackson's Hobbit films add a layer of confusion to this. Not only are they prequels to a film trilogy that had been made without any prequels in mind, but they're based on a book that was written without any sequels in mind -- sequels which were eventually written and adapted into the Hobbit films' predecessors. It's the book-to-movie equivalent of the "chicken or the egg" question, and this leads many to wonder if the Hobbit and Lord of the Rings film trilogies can really function as one six-film series. Can future audiences view the films in chronological order without confusion, or are they better off viewing the films in the order that they were made?
There are a few inconsistencies between the two trilogies, but not many, so let's get those out of the way first.
Even though Ian Holm was available to reprise his role for The Hobbit, playing a younger and more able-bodied Bilbo would have been too difficult for him, so Martin Freeman was brought in. Holm still plays the older Bilbo at the beginning and end of the Hobbit trilogy, and while this fits with the way he looks in the Lord of the Rings trilogy, it does create a bit of a plothole.
See, one of the powers that the One Ring possesses is immortality, which prevents its bearer from aging for as long as they carry it. This is said to be the case with Bilbo in The Fellowship of the Ring, but in the Hobbit films, he clearly doesn't look the same age sixty years after finding the Ring. In fact, if you listen closely to the dialogue from Fellowship that's replayed at the end of The Battle of the Five Armies, Gandalf's line about how Bilbo hasn't "aged a day" is omitted from it.
The argument could be made that the Ring in the movies doesn't stop the aging process so much as it slows it down; after all, Gollum's appearance doesn't stay the same throughout his time with the Ring. However, it's still apparent that having Bilbo played by two different actors was not in the equation at the time that The Lord of the Rings was filmed. This is evidenced by the fact that Ian Holm does play the younger Bilbo in a flashback at the beginning of Fellowship, where we see him finding the Ring.
This leads to some other minor issues. Not only does that version of the younger Bilbo look and dress differently from the one in the Hobbit films, but the entire riddle game with Gollum is removed in that flashback. The scene jumps right from Bilbo picking up the Ring to Gollum screaming that his "Precious" is lost, something that Gollum doesn't realize until after the riddle game concludes in An Unexpected Journey. Future audiences will probably be able to tell that both scenes are depicting the same plot point, but the inconsistencies between them are bound to be a little jarring.
Gollum is another character who differs between the trilogies. Since Peter Jackson's team was still perfecting CGI motion capture in the early 2000's, Gollum was mostly kept out of sight in Fellowship. By 2011 though, his scenes were pretty easy to shoot and render. As a result, people who watch the six films in chronological order will get a Hobbit film where Gollum is shown as clear as day, a Lord of the Rings film where he's suddenly shrouded in mystery, and then two more Lord of the Rings films where he's plainly visible again.
This doesn't really detract from watching the films in that order, though. It just slightly lessens the impact of Gollum's reveal in The Two Towers and makes it easier for viewers to tell which order the two trilogies were made in. Gollum's buildup in The Lord of the Rings might even still work, since viewers are given two whole films to forget what he looks like in between his appearances in An Unexpected Journey and The Fellowship of the Ring.
There are other things that can play out strangely to viewers, especially when watching the Extended Editions. It's odd to hear Bilbo calling for Frodo in Bag End at the beginning of Fellowship's Extended Edition when he clearly saw Frodo leave Bag End at the beginning of Journey, and it's ironic that viewers don't get a clear explanation of what a hobbit is until after the Hobbit trilogy is over. A lot of major characters like Radagast, Thranduil, and Tauriel disappear with no explanation halfway through the series, and no mention of Balin's conquest of Moria is given prior to the Fellowship's decision to go there in The Lord of the Rings. People who forget what Moria is from all the way back in Journey might get confused about this plot thread. Once again though, these are minor issues.
With that said, let's discuss how the two trilogies do flow into each other.
The Hobbit's mentions of characters from The Lord of the Rings further help to bridge the gap between the film trilogies. Some feel that those mentions can be forced at times, but they do help to establish those characters' relevance in the long run. In case we forget who the minor character Gloin is by the time Gimli is introduced as "Gimli, son of Gloin" in Fellowship, we might remember Gimli's name from its prominent mention in The Desolation of Smaug; in case it seems odd for Legolas to be such good friends with a non-elf like Strider in Fellowship after watching the Hobbit trilogy, we'll remember his father telling him to seek out Strider at the end of The Battle of the Five Armies. Screenplays have to be a lot more tight-knit than books, so it's usually good practice for films to elude to story elements that way.
Lastly, it helps that the Hobbit films get darker and more serious to match the tone of The Lord of the Rings as they go on. This was partly by design and partly by necessity, since the Battle of the Five Armies and all of the major character deaths that result from it really couldn't have happened offscreen in the films like it did in the book. Because of this, the Hobbit films give us a comfortable transition from their light-hearted source material to the more grisly Lord of the Rings films.
And that brings us to our final verdict. At the end of the day, it's pretty obvious that the Hobbit film trilogy, while having its own identity, was meant to be viewed in order with the Lord of the Rings and not as a stand-alone series. The inconsistencies between the trilogies are all fairly minor, and they cease to be a problem roughly halfway through The Fellowship of the Ring.
It's doubtful that Peter Jackson will ever go back and make changes to Fellowship to better match it with The Hobbit, and most fans of the series likely wouldn't want him to anyway. And really, he shouldn't. As far as prequels go, the Hobbit trilogy is a strong followup that ties in very well with its predecessors, and its shortcomings in that regard should be left intact just to highlight how few of them there really are. If the perfect prequel is unachievable, then the Hobbit films are three of the closer attempts at reaching that.