Answer:
Romantic and Classical refer to different periods in music history. While the term "classical" often concerns the whole genre of symphonic music, the classical period is loosely is identified with the years 1750 - 1830. The big champions of that era would be Mozart, Haydn, ending with Beethoven.
The dates of when the eras of music begin and end can vary depending on who you ask. Like most of history, then new era still tends to contain the best of the old era. The Romantic era began to see its emergence around the late Beethoven period post 1815. The characteristics of "romantic" music saw more dissonance, longer melodies and a departure from traditional musical structure. All though Wagner wasn't known for his symphonies, his operas may be the best example of a new era in music. Composers like Brahms, Berlioz, Dvorak, Tchaikovsky and my personal favorite Mahler, all show a breaking of the classical era traditions in symphonic form. With the death of Mahler and the rise of atonality and french impressionism, 1910 is the generally when the Romantic era is concluded and the 20th century music takes over.
Classical Symphony examples: Mozart's 40th, Haydn's 104th "London", Beethoven's 1st
Classical/Romantic examples: Beethoven's 9th, Brahms' 1st.
Romantic Symphony era: Berlioz "Fantastique", Franck's "D minor", Tchaikovsky's 6th, Mahler's 1st
There is plenty of debate as to which composer is best identified with each era and the individual symphonies may have their own placement on each side of the eras. Beethoven is the best example because his music shows clear development toward romanticism. While Beethoven's sound can be found in all his music, the difference between his early and late works seem to completely fill in any gap that could exists between Classical and Romantic.
More specifically, in the romantic era, the roll of music patronage started to become more privatized rather than royal courts of the state having a such a heavy influence over composers with their patronage. More experimenting mean new ideas being attempted particularly with instrumentation. Brass instruments now began to play melodies. The dynamics of the orchestra grew with louder louds, and softer softs. While romantic era doesn't actually translate to the the music containing more "love," passion is now a central motivation in the composition of new music in that era.