That's tough because I would say that French Symbolism is one branch of the much more broad category of Modernism. Modernism covers not only a historical period (roughly from early 19th century up until post WWII or the 1960s depending on who you ask). Modernism as a cultural and artistic movement, in my opinion, includes many movements: Dada, Surrealism, Futurism, Fluxus, Happenings, Pop art and Symbolism to name a few. Modernism was all about the avant-garde, all about rejecting tradition in favor of anything new. Certainly revolutionary philosophies like Marxism and Feminism were part of this grand scheme of rejecting tradition (and the Enlightenment's claim of humanistic progress; we got the technological part down.) Anything that pointed towards the future. Anything progressive.
Although Symbolism is a movement which occurred during Modernism, you can still compare the two. Symbolism, like Modernism, was an attempt to get away from Realism and Naturalism in favor of a more abstract artform. The Symbolists sought to use metaphoric language and images which symbolized the essence of being. This essence was also beyond the scope of the senses, so the senses themselves were mere symbolizers of being. In general, Symbolism was more abstract and imagistic than its antagonistic artforms: namely realism and naturalism.
Ok. One Big Difference: Modernism focused on the alienation of the individual in the fast-paced technological world. Some modernists ignored technological advances in their work (except for Marinetti) as it was part of the false Enlightenment project. So a lot of Modernist work was about the anonymity and loneliness of the individual being pushed along passively, zombie-like as just a cog of the masses in the machine of technological progress. Symbolism focused on the subjectivity of the individual and how to evoke abstract meaning by observing everyday images and senses. Although different from Romanticism, Symbolism was similar in its focus on the evocation of meaning through individual experience. So, it wasn't the alienation of the individual or the individual's reaction to the external world(Modernism); it was the subjective reaction, the internal, emotional imagination(Symbolism).