Programming in C will give you much greater control over the device and your programs. It's probably more difficult as a beginner, particularly if you have little or no programming experience.
If you are just playing around with the ESP8266 or doing very simple projects and have limited programming experience then Lua is probably the way to go. If you are doing anything complex and either know C already or would like to develop a knowledge of a widely used language, C would be the better choice.
I haven't used Lua in anger but I've read enough about its limitations to put me off and since I already know C well, that was the obvious choice. The ability to program fluently in C is a useful and transferrable skill. Lua, not so much.