Baroque music is a curious beast. What is usually thought of as "Baroque" is the late, or high Baroque. The big change from the Renaissance to the Baroque actually was quite different from the music of Bach and Vivaldi.
In general, the defining development was Opera. Various theorists and musicians came up with the idea (supposedly based on classical Greek philosophy) that music could be more effective and emotional if it were made both more simple and more free. The complex polyphony of the late Renaissance (Palestrina, Lassus, and Byrd for example) was replaced by a single singer and a continuo line. This continuo is basically a bass line with numbers and symbols to indicate chords. This kind of accompaniment allows a great deal of freedom to both performers (or more, since the continuo line can, and frequently does, involve many players). This freedom extended from tempo and rhythm alterations to ornamentation.
The most clear example of this new style is recitative, where the line between speech and singing is blurred. A good performance of recit essentially involves the singer speaking their part with the drama of a good actor, while at the same time hitting pitches and occasionally letting fly with a barrage of actual music. It's important to note that the short, bland recitatives from Handel's time are quite different from the earlier sort. In an opera like Monteverdi's Orfeo, the recitative is frequently as elaborate and melodic as the arias, and the vast majority of the opera is given over to the recitative.
Some of my favourite aspects of this early baroque music are ones that are sadly lacking from nearly all modern music. Principally among these is the freedom given performers - rhythmic notation was at best, imprecise, because precision wasn't needed or wanted. Many composers gave specific instructions to change the rhythms as one felt fit, or to disregard them completely. Some styles, like french unmeasured preludes were usually written with no rhythm at all - just a series of circles on the staff.
Of course, the Baroque does include such wonderful music as that of Bach, Vivaldi, and Francois Couperin, and they shouldn't be ignored. But I think the earlier composers, Frescobaldi, Louis Couperin (uncle to Francois Couperin), Monteverdi, Castello and others, should also be appreciated at least as much. I expect the main reason that they aren't as well known as the others is simply that their music is much more difficult to perform without a strong understanding of the style of the music - Bach can work moderately well on almost any instrument, in almost any way of playing. But these earlier sounds, with their emphasis on rhythmic freedom, rather than theoretical complexity, require some more specialized work to make them understood.