Neoclassicism was a movement which rose between the period of approximately 1760-1850. The movement was one which finds its originations in Rome, popularized by the writer Johann Joachim Winckelmann, following the rediscovering of the ancient Roman cities of Pompeii and the Herculaneum. It was Winckelmann, in fact, who coined the now often spoken of notion of "noble simplicity" which he made with reference to the classical art of the ancient Greeks, for he admired the "calm grandeur" of Greek art, and felt that it should serve as a template for our own approach to art and design.
The neoclassical style set itself in opposition to the more exuberant Rococo style that immediately preceded it by prioritizing order, symmetry and greater simplicity in design than its Rococo counterpart and one of the areas this movement excelled in, besides architecture, was in the area of the decorative arts -- of which vestments would fall under. If you were to compare these designs to their Baroque or Rococo counterparts, one would note that overall they are much more 'staid' (relatively speaking, though certainly not compared to modernist minimalism).
The designs of neoclassically inspired vestments pick up on neoclassical decorative themes that were borrowed from the classical Greek world, particularly incorporating design elements such as laurel wreaths, festoons (i.e. garlands hanging down from two points) medallions, arabesque designs (i.e. rhythmic linear patterns of foliage and the like), vases, and so forth.
