The Petzval lens design was revolutionary, because it was the first commercial lens with a fast aperture. Previously, the Chevalier lenses had a fixed aperture of f/15, and when combined with recording media with a sensitivity equivalent to a small fraction of today's ISO 1, that meant very long exposures.

Chevalier improved on this with an f/5.6 design, but that suffered from poor sharpness. Petzval's design allowed apertures up to f/3.6, and had great sharpness in the center, making it a much better tool for portraits.

So, that's interesting historically. If this aspect interests you, there's a fantastic and detailed article at antiquecameras.net, with vintage advertisements and pictures of several vintages lenses, although unfortunately not many samples. You can find samples in this Large Format Photography forum thread.

Additionally, the design has very dramatic field curvature, which tends to render the background as a dramatic and distinctive swirl — the "Petzval swirl". Like all such optical effects, this is hard to replicate exactly with a filter without a 3D model of the scene, and if you like the look and are fascinated with the low-tech, lo-fi aesthetic that the Lomography company (and others like Lensbaby) have made their bread and butter, this may be something you'd like to add to your creative options.