What they all said ... +
I'd suggest as a simple definition of sorts "tone mapping is essentially adjusting parameters* in selected parts of your image differently so that you achieve a result which you consider enhances the quality of the resultant image". * Where "parameters" is usually limited to things like contrast, saturation, "curves" and similar - maybe also colour balance (see below) as opposed to additions or subtractions of actual image content). (That needs more "definitions, but you get the idea). So -
Note that any of the following tends to involve "art" and the concept of "better" is, literally, in the eye of the beholder:
Tone mapping can be carried out after the event whereas HDR needs a prior decision.
Tone mapping uses a single image whereas HDR needs 2 or more so TM is unaffected by subjects which change between 'frames'.
Tone mapping can if desired be used to make image changes which do not reflect those seen by the eye in the original but which are felt to enhance the end result. eg colours in a portion of a scene may have been rendered differently than in the body by a change in lighting which affects the image in an undesired manner. This may be sun/shadow, flash/incandescent etc - by "re colour balancing" portions the end effect may be "better".
When carried out in-camera Tone mapping appears to take less processing power (see below).
When shooting RAW + JPG with in-camera the RAW image can be provided untouched and the JPG tone-mapped (this is Sony practice). This makes sense as tone mapped RAW <> RAW.
HDR RAW does not make sense by definition. But to shoot 3-frame-HDR + RAW + JPG would require at least 4 frames of output (3 x RAW + HDR-JPG).
Some recent Sony DSLR cameras provide the equivalent of both tone mapping and HDR options in-camera, with a degree of user control of the degree of effect achieved.
In the case of HDR Sony offer 3 frames with from +/- 1 to +/- 5 EV spread or Auto. They provide output of one "correctly exposed" and one HDR frame. Even when set to 5 EV the HDR results seems mild compared to what I see elsewhere. Effects are useful but not usually "garish".
Sony in-camera tone mapping is termed DRO (Dynamic Range Optimiser). They offer 5 levels + Auto. High levels of in-camera DRO offer near miraculous results in situations where results would have been poor usually so that quality loss due to TM / DRO is potentially acceptable. eg flash photography in a large room will result in vastly more even lighting at extended distances from the flash without utterly wiping out foreground. Needless to say, the more a lowly lit area is "lifted" the noisier it gets, but the results are very useful in situations where no superior or accessible alternative exists.
Care must be taken! Blacks of any depth stop being black and become variants of very deep grey.
At maximum level under flash Samoan faces tend to look like slabs of coffee coloured skin. Ask me how I know:-). I'm "an old white guy", but I have a number of Samoan friends and rapidly found that using DRO level 5 with flash was a bad idea when they were in the photo. At this level the DRO process seems to make an "it's all much the same" decision and produces consistent somewhat monochrome results for large areas of facial skin. I have not noticed that the result as pronounced with other skin tonings, but I also tend to use DRO / TM with care in situations with similar potential.