In terms of how much respect Japan got in the eyes of the world powers, the war acted as a major shift in how westerners saw Japan. For Great Britain, it verified that Japan was a strong ally, able to help protect British interests in the Pacific, while France saw Japan as further complicating their own imperial ambitions in Asia. The most drastic shift, however, is how the U.S. saw Japan, since the victory over the Russians demonstrated that Japan could now be considered a threat to America's new colonies in the Philippines and Hawaii. Before the war, American's thought of Japan as a puppet nation that would support American trade in Asia. After the war proved that Japan could beat a technological great power one-on-one, the U.S. feared that Japan could no longer be easily controlled, and their relations chilled from a partnership to a rival relationship.
On top of this, Japan's victory over the Russians threw into question notions on white supremacy in the imperialist context. Japan proved that technological and military supremacy were not necessarily linked with race. This caused a crisis in the west, since imperialist motivations were based on the idea that non-whites couldn't run a modern country. This caused groups that had previously supported Japanese modernity in the late 19th century to now viewed the Japanese with suspicion after the Russo-Japanese war.
The war represents a turning point, where Japan was given a lot more respect by the west but also became far more feared, leading to a chilled relationship between the west and Japan. This chilling of relationships lead to Japan being treated unfairly in the 1924 Washington naval conference, setting Japan and the west down the path of inevitable conflict by 1938.
Although it only covers it briefly, Theodore Rex does talk about TR's peacemaking efforts between the two and how they were hampered by Russia wanting to save face (Japan felt that need as well, even if they were winning). The Russians clearly lost, except Tsar Nicholas II was reluctant to admit it -- trying to get concessions at the Portsmouth talks were hard to do. (You have to feel bad for Sergei Witte who was representing the Russians.)
What makes this whole time period more remarkable was that the secretary of state, Hay, passed away shortly before all of this and that the president was also trying to keep tempers between France and Germany cool -- Wilhelm II was upset about an earlier incident involving Morocco.
How did the war begin?
I thought I would take a look at this from the European perspective. I have to recommend always Orlando Figes' A People's Tragedy regarding this entire era of Russian history. Russia went into revolution in 1905 for a variety of reasons, but one of the greatest symbols of the 1905 revolution was the failed naval mutiny on the battleship Potemkin prompted by the low morale in the Russian navy resulting from the Russo-Japanese war. Naval mutinies would become something of a "thing" for later revolutions on the left, with red sailors in the Russian Revolution and the Kiel Mutiny in Germany (but see also the Kronstadt rebellion against the Bolsheviks).
More immediately, in 1908, Austria-Hungary annexed Bosnia in a fait accompli. Russia, being in a perilous military state due to the 1905 war, was unable to push back in any meaningful way. Had the balance of power not been altered due to the 1905 revolution and Russo-Japanese war, Austria may have not made such an aggressive move.
The annexation of Bosnia would obviously have very significant ramifications for 20th century Europe and the world.