This question approaches religious dimensions, with firm beliefs being held on every side. The same arguments that are presented for other string instruments also apply here.
For players of baroque to classical repertoire, an instrument from the period in which the music played is an advantage in itself. Some of the workmanship particularly with carving ornaments and inlays are rarely seen on modern instruments. Older instruments tend to be more light-weight.
Many instruments are not all original. Often the neck and/or pegpox have been altered, as many violas d'amore had been converted to violas. Violas d'amore suffer often from neglect, and small problems that could have been easily fixed early on, have developed into major issues. Wormholes are common occurrences in old instruments.
Copies of old instruments can be readily obtained at a fraction of the price. Quality materials in the hands of a capable luthier can ensure problem free enjoyment of the instrument.
Instruments tend to be heavier. One does not know yet how the instrument will develop over time, both in terms of build quality, and in terms of sound.