There should have been an antenna in the box with the motherboard. It's on the spec sheet as "1 x ASUS 2T2R dual band Wi-Fi moving antennas (Wi-Fi 802.11a/b/g/n/ac compliant)" in the accessories section.
Yes, the antenna is mandatory for proper operation of the WiFi card. The WiFi card is inside an RF shield and only has the two small antenna connectors outside the shield, so it's very difficult for radio signals to get in and out of the card without an antenna attached.
Ethernet is almost always better for systems which don't move around often and where it's reasonable / practical to run a cable to them from the router. Ethernet offers predictable and reliable performance with the lowest latency, and it mostly just works without any external factors degrading it.
If you do need / want to use WiFi, the performance from the motherboard WiFi should be excellent with a suitable antenna attached. As long as the antenna is good and not blocked by other stuff, it should perform better than the WiFi in most laptops and phones due to having a proper big antenna. Latency will always be higher than Ethernet, and it's always subject to the general issues that come with any radio communication in terms of interference and signal attenuation. WiFi is also shared bandwidth and half duplex, so you're sharing the theoretical bandwidth with all other WiFi devices on the same frequency (including devices on other networks) and other communications using the same unlicensed frequency bands.
