I've read the first six or so novels, covering up to the end of the OVAs for the first anime series. [Note: now I've read them all, so I've edited to reflect that.] There aren't any major plot differences between the novels and anime, but there is lots of extra material in the novels that could be interesting for big fans of the anime. I'll spoiler-tag major stuff. Some of this material is also in the manga, which has an official US release, although I'm not too familiar with the manga, so I can't say how much.
The first few novels follow the same plot arc as the anime, with Kyousuke discovering Kirino's otaku secrets and helping her meet Saori and Kuroneko, while also dealing with their parents and Ayase. The main difference is that the novels are narrated in the first person by Kyousuke, so we get to hear his reasons for some of his more inexplicable actions (for example, his screaming bout with his father that was in Episode 3 of the anime was, in the novels, depicted as a product of utter terror and desperation, whereas it looked sort of calculated in the anime).
There are a few extra stories in these novels that weren't adapted to anime, like one where Kyousuke buys Manami a present. The first major difference is the novel-writing storyline, which is almost completely different between the novels and the anime. In the novels,
the woman who was Kirino's editor in the anime is actually a failed novelist herself, who steals Kirino's work and passes it off as her own. Kyousuke and Kuroneko infiltrate the publisher using Saori's connections and expose the plagiarism, after which she becomes Kirino's editor as in the anime.
Kirino's novel is never turned into anime as it was in the anime series. Kuroneko has a much bigger part in this story than in the corresponding anime story, which I enjoyed as a Kuroneko fan. The earlier Christmas date story also has a slightly different emotional payoff than in the anime.
In the novels' equivalent to Series I, Episode 11, Kuroneko actually finishes showing her picture drama about Kyousuke and Kirino's relationship. Not only is it one of the funniest scenes in the novels, it also explains Kuroneko's behavior towards Manami after she starts attending Kyousuke's school.
The anime's "good" ending doesn't happen in the novels; only the "true" ending, where Kirino flies off to the United States, happens in the novels.
I have now read the novels that cover the second series, and the Sakurai subplot, a major subplot explaining
why Kyousuke changed to his current "take it easy" persona, causing Kirino and Manami's confrontation and Kirino's hatred for Manami, as shown in Episode 13 of Series II
was not adapted to the anime. This subplot takes up most of the eleventh novel and contains a scene of temporary peace between
Kirino and Manami, wherein Kirino even helps out at Manami's family's shop.
During this scene, Kyousuke recounts how when he and Manami were in elementary school, he
had a meddlesome, class president-like personality that led him to expend a bunch of energy getting a delinquent and nascent otaku, Sakurai, who of course is another cute girl otaku, to come to class. Kyousuke convinced Sakurai to come on their class trip, where he led her up a dangerous mountain peak. Sakurai fell off and was seriously injured; her parents blamed Kyousuke and moved her to a different school, and they didn't see each other again, until Kyousuke seeks her out to apologize in the present. After this incident, Manami convinced Kyousuke to stop being so meddlesome.
This led to the incidents between Kirino and Manami, and later Kirino and Kyousuke, that we see from Kirino's perspective in Series II, Episode 13—which is not in the novels.
The events it covers are implied, but never explicitly shown; in particular, the anime offers a definitive explanation of Kirino's love for little sister eroge, which we didn't get in the novels as far as I can remember.
It's debatable how much you're really missing by not getting the subplot from the novels; I found it to be a huge, bloated digression that added very little to the anime's much shorter overview of the events, and it
adds yet another confession to the string of love confessions that Kyousuke rejects due to his relationship with Kirino, when Sakurai confesses to Kyousuke sometime between his confessions from Ayase and Kanako.
Disappointingly (especially for Kuroneko fans), the controversial ending of the anime is an exact adaptation of the novels' ending, so don't expect the novels to save you from that disappointment. (In fact, it's weird how many of the anime's worst moments are direct adaptations of the novel.)
In general, the side characters are integrated into the novel more thoroughly and relationships are explored at greater depth. We get little mentions of things like the relationship between Kyousuke and Rock (Manami's little brother), or between Kuroneko and Ayase. There are also some side stories that shed extra light on certain characters (including Hinata Gokou, Kuroneko's middle sister; Kouhei and Sena Akagi; and Mikagami, who posed as Kirino's fake boyfriend in Series II) and show us things that happened offscreen in the anime, or outside the main timeframe. There's even a crossover story between Ore Imo and To Aru Kagaku no Railgun, where Kirino and Mikoto meet on a talk show while Kyousuke and Touma Kamijou hang out backstage. There's a lot of good humor and interesting exploration of the characters in this extra material.
While reading the novels, it's pretty easy to see why all of these things were cut from the anime; the Sakurai subplot introduces a not-very-significant new character at the last minute, and I found the anime version of the main plot perfectly intelligible without it. The rest has nothing to do with the main plot, and is mostly of interest to hardcore fans. I wouldn't say you're not missing anything by skipping the novels, but you're not missing anything essential—you can get about 90% of the Ore Imo experience, including 99% of the bad parts, with just the anime.