Both offer a wide range of products, from entry-level to high-end, and both have a good reputation for quality. In this case, everything is up to your budget and according to your needs. If you're looking for the best possible performance, you'll want to spend more money on a high-end card from either ASUS or Gigabyte.
However, if you're on a tight budget, you'll likely be happy with a mid-range card from either brand. Generally speaking, ASUS cards are known for being slightly faster than Gigabyte cards, but both brands offer excellent performance.