Narrow bandwidth.
Susceptibility to tuning/frequency shift.
But there are techniques for broadbanding them. You want the lowest permittivity you can have-- basically, you want air. Also, there are structures for broadbanding the feed. Check out "CAD of Microstrip Antennas for Wireless Applications" by Robert A. Sainati. And there are some newer books on it.
34-year RF & electronics engineer.