A tropical storm requires heat - from an ocean at above 26 degrees celsius surface temperature. When it passes over cooler water or land that energy supply disappears, and so its internal structure collapses. If caught by mid-latitude winds (e.g. Atlantic westerlies) it can morph into a standard low pressure system (e.g. Hurricane Oscar that was still intact when it brushed the UK and Iceland as a strong low pressure system).
What that change is called and how it works I don't know.
Yes, a very good question.
I too am not an expert in the definition of storms. What I believe I have learned is this:
Hurricanes and Typhoons (differing in name only because of the region in which they formed) have warm cores with uprising air. These occur when sea surface temperatures exceed some critical threshold, often taken as being 26 degrees C. Much more is involved though to drive their formation and destruction. High altitude shear plays a strongly limiting role.
Extratropical storms form in the northern and southern oceans where the sea surface temperatures are lower. These systems have cold down flowing cores making them different from their tropical cousins. Visually, they comprise alternating white cloud and dark (open air) bands wrapped around a central area, as contrasting from tropical systems that are solid white clouds in bands wrapped around the core. They are typically much larger in area than their trivial cousins. And though their wind speeds tend to be lower and their central pressures higher than their tropical cousins, they can rival them in each of these areas. Tropical systems have a true eye. Extratropical systems may have a loose eye, but generally do not.
Hurricanes and Tropical storms that form and move poleward often either convert from warm core to cold core systems once the sea surface temperature drops below some critical threshold. They also quite often merge with extratropical lows creating even larger and more powerful systems in the process. In much rarer cases, systems can and have apparently moved toward the equator and converted from extratropical nature to hurricane or typhoon nature.
I suspect, but do not know, that there may be a difference in the atmospheric driving forces as well. Coriolis forces seem to play a large role with tropical systems. High altitude circulation appears to play a large role with extratropical systems; both in their creation and in their destruction.
If these are true, the three band atmospheric circulation no doubt plays several roles in driving storms, and in forcing the conversion of tropical systems into extratropical ones. With the failure the three band circulation, this may change allowing tropical systems to range further from the equator.