Shelley followed much in way of Romantic tendencies in his work. Yet, I also believe that there were some distinctly different elements in his writing. Whereas thinkers like Byron and Wordsworth were animated by the expression of Romanticism in the present, there was often a condition of the future tense invoked in Shelley's writing. He is concerned with his place in poetry, how he will be perceived, and how art has the capacity to last longer than more human endeavors. For example, in the poem "Ozymandias," the theme is to explore the immortality of human deeds and actions. While the individual will be reduced to dust and ashes, the question that emerges is what does last. In "Ode to the West Wind," the probing issue throughout the poem is how Shelley, as a poet, will gain immortality and how will his work live on even when he does not.
A very important thing to know about Shelley is that he was an athiest. This immediately changes how he saw the world and how he expressed his poetic vision. While religion shouldn't be the basis for judging a poem, in this particular case it is a vital aspect. Think of William Blake, an early Romantic, whose work was substantially Christian-based and Biblically influenced. His beliefs are not only found in his work, but they actually create his work and its message.
With Shelley, then, we see someone who does not believe that God is in nature, or that Christ is in a flower, but as someone who views life as a fleeting thing. This is beautiful in its own right. It approaches the Romantic's favorite subject, nature, with new eyes. Is God in nature, is nature God, or is nature simply a beautiful accident?
Shelley's work deals greatly with nature and natural imagery. This is true for all of the great Romantics. Shelley's beliefs on what nature is, and how it came to exist, steer away from religion. That is probably the most concise way I can express the difference between Shelley and other Romantics.
Consider what Romanticism is: nature v. culture, with culture being negative and nature being positive. There is a contradiction: one cannot write about nature without first having culture; however, this is mostly overlooked when analysing Romantic poetry. Romanticism is also about the outsider v. the insider, with the outsider being the ideal human. Poets such as Lord Byron (Shelley's best friend) break that rule by being of a noble class, while poets such as John Keats (another close friend of Shelley) embrace it with their lower class backrounds. Shelley automatically becomes an outsider due to his lack of religion: a very serious issue in the 1800s.
I encourage you to browse through the victorianweb's Shelley page, as well as glance through Shelley's Prometheus Unbound. Both will give you further insight into this great poet.
I hope this was of help. Good luck!