Hapkido and Aikido. Where to begin? It is indeed true that the Chinese script for both arts is the same, though they developed independently of eachother during a point in which there was some sort of trade ban between Korea and Japan. One branch, upon discovering that there was a style in Japan which shared its written named, changed its name to simply Kido, but you don't see that term around very often.

Aikido and Hapkido both stem from Daito-ryu Aikijujutsu, and the same teacher. Just how GM Choi learned aikijujutsu is a matter of some debate, but he did indeed learn it, though his footwork was very different (some have theorized that he learned the art by watching only, and that the students' hakamas concealed view of the footwork, though this theory is iffy).

Choi's students added movements from Taekkyon (Korean folk-MA) including more footwork and an extensive array of kicking techniques. Taekkyon was also one of the sources for TKD's kicks. While TKD went bouncy and flashy with their kicks, most of Hapkido's "trademark" leg techniques are very hard-hitting, though somewhat telegraphed, emphasising full-body power (though many still sorely lacking any sort of root). Most Hapkido people show off with their tumbling and breakfalls rather than their kicks.

After some extensive politicking between branches, what you see today in Korea and some schools in the west (the ones who don't just use it as a supplement for their TKD) can vary, but is pretty distinct if you know what you're looking for. Most branches still bear a strong resemblance to Aikijujutsu, but only the IHF (Korea's IHF, not to be confused with Bong Soo Han's IHF) defensive method (Hankido) bears any resemblance to Aikido, as our founder GM Myung Jae Nam was highly accomplished in both Hapkido and Aikido. The Hankido method looks a lot like Aikido, but the power generation is very different. They're even more hardcore about full body kicks and punches, and all techniques involve circular force.

Hope that helped.