Wi-Fi versus Z-Wave & Zigbee
Wi-Fi was originally designed to cover large areas and facilitate high-speed data communication (such as media streaming). Wi-Fi’s drawback by being able to facilitate high-speed data communication is that it is a power hungry radio frequency, which makes it a sub-optimal radio frequency for battery-operated devices, such as door locks, scales and sensors.
Radio frequencies such as Z-Wave and Zigbee were also designed to cover large areas, but, in contrast to Wi-Fi, highly power efficient, which makes them ‘perfect’ radio frequencies for battery-operated devices.
Z-Wave versus Zigbee
From a consumer's point of view, the most important difference you should be aware of with regards to Z-Wave and Zigbee at the time of writing the latest update to this answer (Mon, Nov 18, 2014) is that products which use a Z-Wave radio frequency have a higher likelihood of being able to pair with other Z-Wave products compared with Zigbee products.
The above means that the likelihood that a Z-Wave product from a Company A can pair with a Z-Wave product from a Company B is higher than a Zigbee product from a Company C being able to pair with a Zigbee product from a Company D.
If you are considering to buy Z-Wave products it is important to know that Z-Wave uses different radio frequencies around the globe, for example 908 MHz in the United States and 868 MHz in Europe. So if you are living in the United States and use a Gateway/Hub with the 908 MHz frequency and buy a Z-Wave product from Europe with the 868 MHz, your US Smart Home Gateway/Hub will unfortunately not be able to pair with your EU Smart Home Product.
The key reason why Z-Wave products overall are more compatible with one another compared with Zigbee products is the fact that Zigbee is an open standard which is (unfortunately) often tweaked by Zigbee product producers to realize a closed ecosystem in which only the company's own products can pair with one another. Z-Wave on the other hand is a locked standard with fixed interoperability rules which cannot be tweaked by Z-Wave product producers, resulting in a significantly more user-friendly ecosystem for consumers from a product compatibility point of view.