They both operate in the unlicensed wireless bands. However their applications and specs are different.
In North America, ZigBee operates in the 2.4 GHz frequency range and has a maximum speed of 250 kilobits per second (kbits/s). In Europe it uses 868 MHz at 20 kbits/s. It uses a very low power rating of only 1 milli Watt (1 mW), 0 dBm, so the range is only about 30 feet (10 meters). It can definitely have and/or cause interference issues with WiFi and bluetooth.
ZigBee has very specific applications such as telecommunications, smart energy, home automation, building automation, remote control, and some retail services.
Z-Wave operates in the 900 MHz wireless spectrum and creates a wireless meshed network. Basically each “node” can talk to other nodes by relaying through adjacent nodes or if they are close enough, they talk direct. You can have up to 232 devices. The maximum speeds are very slow at either 9.6 kbits/s or 40 kbits/s. The signal strength is also only 1 mW (0 dBm). It is possible in open space to go up to 90 feet (30 meters).
Z-Wave is more typical in control and monitoring devices such as door locks, security/smoke sensors, controlling your lights and climate, and other home and small facility applications.
If you go to Home Depot or Lowes and buy some wireless home control products, they are probably using Z-Wave technology even though they don’t state it. It is the simpler of the two and more common. ZigBee is more versatile however much more complex resulting in more difficult development. If you google search either of these, you will find a variety of interesting products available.
The reason I became aware and interested in these technologies is due to a Kickstarter project I came across, the Almond+. This WiFi router is compatible with many Z-Wave and ZigBee sensors allowing you to control them including with your Android or iPhone smartphone.