mjhjmtu

Here in Nmap we have -sS option, which starts a "half-open scan."

 ➜ ~ sudo nmap -sS 192.168.1.4
 Password:
 Starting Nmap 7.70 ( https://nmap.org ) at 2018-07-04 20:23 CST
 Nmap scan report for 192.168.1.4
 Host is up (0.000082s latency).
 Not shown: 999 closed ports
 PORT STATE SERVICE
 49159/tcp open unknown

then I use -sT option, which start a "full connection"

 ➜ ~ sudo nmap -sT 192.168.1.4
 Starting Nmap 7.70 ( https://nmap.org ) at 2018-07-04 20:23 CST
 Nmap scan report for 192.168.1.4
 Host is up (0.00018s latency).
 Not shown: 999 closed ports
 PORT STATE SERVICE
 49159/tcp open unknown

The results of these two scans are exactly the same.

Question: How does a "half-open" TCP scan work, and can I simply use it instead of "full connection scan"?

The -sT option does a full 3 way handshake. The -sS option is a SYN only scan. It never makes the full connection. A syn scan waits for a SYN/ACK (SA) or a RESET packet. If it gets either it knows how to mark the connection. The -sT option performs a full connection waiting for the S/SA/A flags.