In my view inverters are best thought of as devices to close the gap between fixed belt or gear speeds giving a seamless speed range. Typically a good device will give you ± 1/3 rd of nominal nameplate speed with little enough variation in effective power output as to be unnoticed in normal use. The better VFD's are engineered to increase drive at lower speeds giving more torque to partially offset the inevitable power loss as speed falls.
Outside the ± 1/3 rd range things start getting compromised. Things still work just fine but you need to be aware of whats going on. In particular at lower speeds power loss becomes noticeable and internal cooling fan efficiency falls drastically so there is a risk over of overheating. Generally inverters are pretty good at only delivering the power the motor needs so the overheating issue isn't as bad as some might fear providing loads are light. Power loss is the main issue as, usually, low speeds are required to deal with large components so the power increase when belt or gears are used to reduce spindle RPM relative to motor speed is very desirable. The speed increasing capability of an inverter can really score here as permitting you to use a lower speed fixed drive setting and run the motor faster getting even more power onto the work. Conversely when you are cutting a short thread, winding a coil or spring or its great to be able to crawl round.
its important to consider back gear speed capabilities. Open gears, as used on most small machines, aren't happy if run too fast. Many makers advise that the highest input belt speed should not be used with back gear.
It has been convincingly argued that best practice is to reduce the countershaft speed for VFD use by fitting a smaller pulley on the motor. Generally high speeds are only needed for small work so power loss when running fast is hardly a worry. Setting up in that manner lets you run in open drive down to rather lower speeds. Which, if nothing else, is quieter than back gear. Biggest problem is getting satisfactory drive from a small Vee pulley on the motor. Which are usually undersized as standard anyway. Multigroove belt systems are much more satisfactory. Suitable belts are readily available these days and pulleys easily made or reasonably priced if you don't fancy the work.
An oft overlooked advantage of VFDs is the very repeatable run down to rest time after hitting the stop button. Correctly programmed with a suitable microswitch arrangement connected to the remote control inputs this can be exploited to make an excellent electrical bed-stop for cutting to threading to a shoulder purposes. At moderate speeds maybe half a spindle turn from switch to stop.
Clive.