How is a light novel different from a regular novel? Except for light novels being... light, are there any other differences?
A light novel is short in page length and usually features intermittent manga-style illustrations --- but that same description can be said of many Japanese children's novels! In contrast, light novels are for all ages & demographics. Although it is approximately equivalent in length to the Western novella, the light novel cannot be called a Japanese novella: the Japanese term for "novella" is 「中編小説」(chuuhen shousetsu), and is not used to refer to light novels (it is used for short stories by Murakami Haruki, for example, which are considered high-quality literature).
Though each published volume is short, light novels can stretch into long series. Some of the series premier in magazines, so you can think of those like literary serials (think Charles Dickens); a serial is episodic, "a printed format by which a single larger work, often a work of narrative fiction, is published in sequential installments." As such, a serial is written differently than a full-length novel that gets published into a chunky book at once. A serial must grab the reader's interest in the first chapter, and plot out the development so that at least one exciting thing happens in each chapter. It intentionally ends chapters with a cliff-hanger when possible. It cannot leisurely build to the climax, or take a whole chapter or so to only tell you what a minor character is doing somewhere else (which a novel printed all at once is free to do). Thus, the format of the story is structured in a specific way that other novels are not bound to. This is another aspect in which the light novel differs from average Japanese children's novels. This is also why, if Harry Potter had been written in Japanese with manga-style illustrations, it still would not have been a light novel: it is not short in length and it was published all at once rather than serialized.
A large percent of light novels are read by people involved in some sort of sub-culture, such as manga, anime, and/or gaming. However, just because a certain percentage of light novels include moe, light novels are not limited to such content and tone; all genres are included, from horror to romance to sci-fi. Some are the novel-equivalent or adaption of shoujo manga and are marketed at young girls. Some genres are less likely to be exported based on the otaku demographics outside of Japan, so those genres are less well-known to the international subculture community.
In contrast to standard Japanese (and Western) novellas and novels for adults, light novels are often illustrated.
Are light novels used exclusively for anime or manga adaptations?
No, some light novels are serialized as their own original works in non-manga magazines and never have a relation to manga.
Some light novels now originate online and never have a relation to manga or any other mediums.
Some light novels are created as adaptions of manga/anime/games after the manga series or game has gained a lot of popularity.
Some light novels start as original works and then get adapted into manga/anime/games/live-action afterward.
Are there any light novels that do not use manga-style drawings?
Yes. Artists of the earliest light novels included those who worked in 油絵 (abura-e = oil painting) and 水彩画 (suisaiga = watercolor). In 1987, the 少女小説 (shoujo shosetsu = girl's novels) genre began, and for the first time shoujo manga style was used, which helped establish the trend of manga-style art as the standard for light novels from the 1990s.
While illustrations are a common feature, some light novels do not include any illustrations whatsoever. The ones without any illustrations are less likely to be exported to other countries and are therefore less well-known outside of Japan.