It is worth adding that the term Violone actually referred to different types of instruments in different countries in different periods. Its history can be traced back to 16th century Italy, where it functioned as the contrabass (16 foot) member of the viol (viola da gamba) family of instruments. It usually had six strings (five in some cases) and was usually tuned an octave below the bass viol.
Though it is commonly thought that the violone is the contrabass' ancestor, it should be noted that 16th century Italian makers such as Amati, Gasparo da Salo and G.P. Magginni made 16-foot instruments following both Violone and Contrabbasso models, the latter being somewhat larger in dimensions, and having three, four or five strings.
It is hard to make a precise judgement on the lineage and development of these instruments, partly because makers were experimenting with different models and widely varying dimensions, a practice which continues even today (this in contrast to the history of the violin or the cello). While the Italian violone was properly a viol, it is my opinion that the contrabass was, and still is, a hybrid, bastardized instrument.
The violone and the modern contrabass have a similar construction, but quite different proportions. The violone usually has a string length of 90 to 100cm, while the average string length of the contrabass is 105cm. The contrabass also has a higher angle to its neck (for greater string tension) which, together with deeper ribs, makes for increased loudness. But perhaps the biggest difference between the violone and the contrabass is in their sound. The sound of the violone relies on resonance (from the six strings) and does not have much volume. The contrabass, in contrast, relies on greater string tension (with a reduced number of strings) and a deeper sound box to produce a bigger sound, at the expense of sympathetic resonance. This also explains the different functions of the instruments. The violone was a continuo instrument, suitable for playing in small ensembles and in small rooms. The contrabass was, however, capable of leading an orchestra (see Domenico Dragonetti) and filling a concert hall with its sound.
Since both these instruments seem to have co-existed in 16th and 17th century Italy, it might be possible to think of the contrabass as a member of the violin family, which originated in street and troubador music of the early renaissance. The contrabass (with three or four strings) was loud enough to be played for dances on the street, and because of its great volume was eventually introduced into orchestras all over Europe. The violin family of instruments (for the sake of this discussion including the contrabass) have finally won over the viol family (including the violone) during the Baroque period with the development of the orchestra and the concert hall.
The violone was still being used in orchestras, even in the time of Haydn and Mozart, but had slowly evolved in the German/Austrian tradition, as can be witnessed in its use in 18th century Vienna. A substantial number of concertos, as well as divertimentos, trio sonatas and other chamber music were written for the violone. The viennese violone had five strings in most cases and had the standard tuning of A-F$-D-AA-FF (note the bottom F-natural!) In some cases the soloist would tune it half a step higher in order to play in E-flat and related keys. But as pallotta says, composers found it boring to have to avoid harmonic modulation.
The 19th century saw the demise of the violone, as well as the rest of the viola da gamba family. The musical world became used to the modern contrabass, which was capable of huge sound and was frequently used for special effects ('the roaring of the cannons', 'the tempest' - as can be seen in late Baroque French music). All this came at the expense of sweetness and clarity of sound. With the disappearance of the viennese virtuoso school, the level of contrabass playing sank to an all-time low, as can be witnessed from the constant ranting of contemporary composers and other musicians, eventually making contrabass players (together with viola players) the butts of jokes in the orchestra.
The 20th century saw the revival of the violone, together with all other early music instruments, and it is used again in authentic performances of renaissance, baroque, and even early classical music.
For more information you can read the excellent, if a bit over-zealous, book by Alfred Planyavsky, The Baroque Double Bass Violone. Another very good book is Paul Brun's New History of the Double Bass.