Basso continuo, the bass system as used in the baroque era, give a distinctive sound to its music. Baroque music is instantly identfiable as such because of this. Although most definitions of basso continuo confine themselves to the technical aspects, it had stylistic characteristics as well. During the baroque era, the sound ideal was a firm yet independent bass line under a highly ornamented treble. The bass line was also often written contrapuntally to the treble, so many times the bass line echoed the melody or vice versa. Most baroque bass lines have melodic interest lacking in later classical music.
To my ears, this combination of characteristics, the firmness of the bass (almost always doubled an octave lower by a double bass), the crispness of the harpsichord (the usual continuo instrument in profane music), combined with the contrapuntal writing, gives baroque music its urgent drive. It's also important to keep in mind that in the baroque era the harmonic rhythm was much faster; chords often changed with every beat, whereas in the Classical and later periods chords typically change once a measure or less.
Two good examples of a typical melodic, yet firm and resolute basso continuo are the third movement of Bach's Brandenburg Concerto No. 2 in F major, in which the bass line is an independent and equal voice in the contrapuntal texture, or Alessandro Scarlatti's aria "Si suoni la tromba," in which the bass echoes the trumpet and vocal parts.