According to Wikipedia, the Trevi Fountain is the "largest Baroque fountain in all of Rome". As a Baroque scholar, I can tell you that this is not entirely true. Originally, Pope Urban VIII hired Bernini, the most famous Baroque sculptor in the world, to build a new fountain on the other side of the piazza so that the Pope could see it out of his window in the Quirinal Palace. Bernini and Pietro da Cortona were responsible for the design of the original Trevi Fountain, but was later torn down and replaced with a different design. Bernini's only lasting impression was the new location in the piazza, while Cortona heavily influenced the layout and design of the fountain. In 1732, Alessandro Galilei won a competition to redesign the fountain. Galilei used Bernini's location, Cortona's Late Baroque classical layout, and incorporated his own Rocco stylistic elements into the iconography. The various depictions of Triton taming the waters and the naturalistic rock formations are Rocco elements, while the overall Classical design comes from the Late Baroque. A triumphal arch in the middle (compare to the Arch of Constantine near the Forum and the Colosseum). The repetition of the tritons and horses adds symmetrical balance to the composition.
The Trevi fountain make absolutely no advancements in art, it is simply a large, obnoxious fountain that makes an average 3,000 euros a day off of the coins tourist throw in. I am not telling you to skip the Trevi Fountain when you visit Rome, because it is beautiful and definitely something you need to see. Just make sure you visit the fountain as an informed tourist!Bernini was most influential and revolutionary Baroque sculptor in the world, after all the Rome is the most important city in the world for Baroque art! Galilei doesn't even make a blip on the Art Historian's radar.
At the top of the steps, the building with the two tall towers behind the obelisk, is a French church. The French diplomat requested the large sum of money be used by the church to build something lavish in the empty space below the church to make it look more important. The architect of the steps is unknown and the project was not completed until around 1730. Today the Spanish Steps are a large gathering place for tourists, street performers, artists, and pick-pocketers. Fontana della Barcaccia, located at the base of the steps, was a collaboration between Lorenzo Bernini and his more famous son, Gian Lorenzo Bernini.