I have read that Liberia was the only independent country of Africa (after Italy's conquest of Abyssinia).
How would the relationship between the US and Liberia differ from ex. any of the dependencies of the British Empire and Britain?
I'm not quite able to address the comparisons to former British Crown dependencies, but I know that the history of Liberia is quite unique, especially with respect to the involvement of the US. I have done some work tangentially related to the American Colonization Society in the early 1800s; I'm fairly confident that I can get to the heart of the question you are asking.
No, Liberia was never a US protectorate - at least in the modern definition of the word.
Americans in the 19th century were grappling with how to deal with the effects of slavery and forged the idea that free blacks would find richer lives if returned to Africa. The American Colonization Society was founded in 1817 upon these ideals of emigration after emancipation. I won't get too in detail about this or a great deal of the nuance involved here because it seems like you are more interested in the evolution of legal status of Liberia.
Beginning in the 1820's, The American Colonization Society sent ships over to the Pepper Coast with volunteer freed blacks. They bought or forced locals off the land, established Monrovia, and began to grow colonies in the region.
By this point, we have a smattering of other societies from Maryland, Virginia, and other states who have formed their own colonies in the area. While there was some funding provided by the US Congress for these expeditions, the colonies remained in the hands of the directors of the societies.
In 1838, many of these societies merged to form the "Commonwealth of Liberia", which was still under the control of the directors of the American Colonial Society. Soon after, the board appoints the first black governor, a Virginian-born freeman named Joseph Jenkins Roberts. The economy is growing and reports from the Society express their pleasure in the developments thus far, "Who can doubt that the experiment of African colonization has been successful?"
However, there were continued difficulties with the British Government as traders failed to understand the international status of the Commonwealth as an independent sovereign entirety. These problems flared in 1846 when the message was sent to the British that "Liberia was not a colony of the US Government," and the British responded by not recognizing the sovereignty of the Commonwealth and instructing merchants to not abide by the Liberian trade duties (Twenty-Ninth Annual Report, p. 26).
Evidence suggests that this conflict with the British trading vessels was a primary cause in the American Colonial Society suggesting that Liberia declare itself independent, which they did in 1847 by forging their own Declaration of Independence and Constitution with JJ Roberts as the first president.
Liberia has been an independent State since. I could speculate as to why Liberia wasn't taken over during the Scramble for Africa, but someone more well versed in the African colonization would have to chime in.
For more information of the early formation of the American Colonization Society:
Early Lee Fox, The American Colonization Society, 1817-1840 (Baltimore: Johns Hopkins Press, 1919).
Old timeline produced by the Library of Congress.
I can pull up some other readings, if people are interested.
De Jure, no. The 1869 treaty of friendship between the United States and Liberia stated that "the United States engages never to interfere in the affairs between the aboriginal inhabitants and the Government of the Republic of Liberia in the jurisdiction and territories of the Republic" (that is, a pledge not to interfere in Liberian internal affairs). Additionally, that same treaty contained no language obliging the United States to guarantee the independence of Liberia from aggression. So, absent any language establishing a policy of protection of Liberia, in the legal sense Liberia was not a protectorate.
De Facto, it might be fair to consider Liberia a quasi-protectorate of the United States. Liberia certainly tried to conduct foreign affairs with Britain and France on her own behalf. But, American diplomats took actions to remind France and Britain of the "peculiar relationshp" between the US and Liberia, and to dissuade those powers from seeking influence in Liberia.
Additionally, a territorial dispute between the Republic of Liberia and France arose in 1884 over establishing a border between French West Africa and Liberia. During this dispute the American ambassador in Paris argued that "the quasi-paternal relationship the United States bore to Liberia authorized the United States to interpose its good offices for the benefit of Liberia".
So, let it suffice to say that for its part, the United States at times treated Liberia as a protected state on the basis of a "recognized relationship", although there was no treaty clause allowing or obligating the United States to do so.
The Map of Africa by Treaty discusses the various diplomatic actions the United States took with regards to Liberia on pages 778-779.
