The only thing I could add to what's pretty much been covered is the popularity of different compositional forms. Forms like the Fugue were popular during the Baroque era as opposed to Symphonies which became more popular during the Classical and Romantic period.

Also, the Romantic era tended to be much more secular.

I think that the easiest way to explain it, to someone with NO formal musical background is in the overall feel. Tell them that Romantic music sounds somewhat like a film soundtrack (sort of), and that Baroque sounds more like a dance in a medieval castle, (not really, but...)

It's the vibe, that the average person who isn't familiar with theory or classical musics is going to respond to. Tell them that Romantic music is flowy and dramatic, Baroque music is rhythmic and hypnotic. Tell them that Romantic music is the Nutcracker, and Baroque music is what Tenacious D starts out with, singing about Bach and Rock.

Of course, the easiest way would be to just play them some music from the corresponding periods...

-Zack

A Pretty Subjective Topic-

There are vast similarities and differences between Baroque and Romantic music. So the discussion can get hairy. I'm not sure we can sum it up in a sentence or two, which is why there are eons of scholars like Christoph Wolf at Harvard writing bibles on the matter.

- I think the biggest difference, would be the influence of the more "balanced and symmetrical" classical music period, whether appreciated or not. Note that I used quotes, and also note that baroque composers were not yet influenced by the classical era. Power-house composers like Brahms had a great appreciation for the symmetrical quality of the classical era and you can see it throughout almost all of his work in comparison to other romantic composers.

* Also, many argue that romantic music is more smoothly interconnected, especially as ideas like "legato" are more emphasized than before (particularly, between movements of the same work), which could also be an influence from classical music and variation.

* Flourishing or embellishment is prevalent in both eras, in their own ways, especially since a lot of research points to the use of vibrato in Romantic music. The Baroque era is the very example of embellishment which we stereotype in any early music.

* It's really a level of evolution, as the Baroque era was also very passionate, but music evolves as society changes.

* I don't think my answer encompasses everything, as what previous people mention about more romantic music being secular, or the rhythmic baroque qualities, the different instrumentation, and tuning systems also show clear differences.

I hope this helps! Cheers!

Beethoven has the perfect balance IMO