For many reasons, Americans have become attracted by "high-speed" rail, and thus the term has been used widely when what is really meant is highER speed. That is, making a normal train go a little bit faster, on a normal route.
Just to muddy things further, in Europe and elsewhere a "high-speed" train (the streamlined kind that are electrified and capable of 200+ mph) often will leave the "high-speed" route alignment and continue the rest of their journey (more slowly) on the regular rail network. Not unlike leaving the interstate and continuing on a US or State highway. In this way, the zoom of the high-speed line is taken advantage of by a wide region, using the fast line for only a portion of the trip.
Like water turning to steam, there is a phase change in the engineering requirements for running trains faster than 100-135 mph or so.
Different kinds of track, route alignment, electrification and equipment are required.
Starting with the track: to maintain speeds above 115 mph or so, rail must be welded. No clicky-clack allowed. Ties must be concrete in order to keep the track structure stiffer and keep to much closer engineering tolerances. There is an advantage in welded rail and concrete ties for slower speed mainlines as well: both are easier to maintain and last longer. The Vermonter route over the New England Central Railroad is being converted to welded rail and stretches of welded rail exist on portions of the Vermont Rail System. Amtrak's Northeast Corridor is an example of concrete ties, but none exist in Vermont.
True high-speed track requires a new alignment. Most of the rail network was laid out in the 1800's, by hand. It's served us well. But to go fast, curves must be slight (less than 3 degrees or so). Some existing lines are more curvy than others and some improvements can be made on existing lines, but nothing over an average speed of 80mph is feasable on existing lines - less in the mountains. Usually (but not always) the new alignment is a passenger only route, designed for lighter weight trains.
Electrification is necessary. Faster than 110mph and the fuel use starts to become prohibative, as does hauling around the weight of a diesel power plant. In addition, to go beyond 135 mph or so, electrification must be the modern constant-tension design (which most of Amtrak's northeast corridor isn't).
Then the trains themselves are different. Besides using electrical power, there is increased attention to areodynamics and weight, which are bigger factors at high-speed. Again, engineering tolerances are tighter.
As long the gauge is the same and the route is electrified (or the locomotives switched for diesels) these trains are fine on the regular rail network as well. Hypothetically a high speed train could run from Washington DC to New York in an hour and a half, continue at high speed to Hartford CT half an hour later or Albany NY an hour later, then switch to the current routes and proceed into Vermont at 100mph. The total journey time would thus be much reduced.