Some of these have been mentioned, others not, or not correctly:
LionChief (no plus) will run on any layout with a constant voltage, such as TMCC,Legacy, and DCS layouts, however they can only be controlled from their own remote control.
LionChief Plus, will also run on any layout with constant power, but also have a switch that will let them run as a conventional locomotive with transformed controlled speed and direction, bell and whistle. You can not fire the electro-couplers however, when using conventional control, or activate the announcement sounds.
LionChief engines have basic forward/reverse control from the remote as well as Whistle, Bell, and announcement sounds, but the over all sound quality is the lowest of any current Lionel products. In truth the sounds are ok, but nothing special.
With LionChief Plus, you gain a somewhat better sound system, that can play several sounds at the same time. You can also adjust the volume of the sound system, and you gain the electro-couplers, which can be opened by 'double-clicking' the whistle or bell buttons on the remote.
The other major features a Plus engine has are a fan-driven smoke unit which will produce a much larger volume of smoke as well as a better looking effect than the puffer units on non-plus engines, and the speed control. LC+ engines will hold at whatever speed they are set to, where as regular LC engines will speed up and slow down as they encounter grades, differing track voltage, curves, and different loads.
There are a number of internal differences in how the electronics function, but I don't think there is a need to get that in depth just yet.
JGL