Both the terms Wi-Fi and Wireless LAN, or WLAN, refer to a wireless networking technology. If a product or service is called specifically Wi-Fi, it has been certified as meeting the Wi-Fi Alliance's Standards.
The Wi-Fi Alliance is a group of some of the largest manufacturers of wireless networking equipment. This organization has agreed on standards for the type of signal and other technologies that their products use to ensure that various network-capable appliances work together. Only products that the Wi-Fi Alliance has tested may label themselves "Wi-Fi Certified."
Wireless LAN is the generic term for the technology used by network-capable devices. There is no guarantee that WLAN devices will use the same type of signal and network protocol as your other wireless devices and therefor no guarantee that the devices will work together.
Though the Wi-Fi alliance has not approved a WLAN device, that does not mean that it cannot use the same signal and protocol as a standardized device. If a WLAN device is less expensive than an Alliance approved device, it could be worth your time to read the specifications to check whether it is compatible with your other equipment.