It's not pretty:
I read a book that followed this idea, only the plants could consume mechanical things and displace them (Sorry, can't remember the name). This is just the stuff I can get a handle on. It would tear apart culture and society, collapse globalism if not industrial civilization, decimate humanity, and completely trash the environment.
Your plants would lead to their own mass-extinction event. Anything that couldn't eat one would starve. If the plants were poisonous, they could directly kill animals. Other plants couldn't compete and would be choked out.
People would retreat to any region these plants didn't grow well in (deserts, for example). Cities can be like deserts, however. Anything people could do to kill the vegetation would be done (viruses, bacteria, herbicides, flamethrowers, etc.) Eventually, people would figure out something that worked and use it, but this could destroy all the plants and lead to yet another ecological collapse. Islands would be refuges, since it would be a small enough area that intense efforts could probably still eradicate the plants (and because of transportation, see below)
- You would need to produce food in greenhouses or very protected fields. If any animals can eat the stuff, people would probably start shepherding these animals for food and to dispose of the vegetation (I like the idea of goats, since they are supposed to be able to eat anything; more likely it would be something like grasshoppers) Much of the human race would still starve to death.
Transportation would collapse, as roads and train lines became impassable. Transport would be by ship, if at all. I assume these plants aren't aquatic, unless some of them are. Inland, air transportation would be critical (assuming you can somehow make something like butanol from superplant) as the area to keep clear would be small.
No transport means no industry, mining, etc.
- The plants would become the chief resource. Dried superplant power plants would generate electricity. If they could be used as building material, everything would be made of them. If you could convert them somehow into ethanol, butanol, or whatever, this would become the fuel that would run any machines.
- People would render the environment around them as inhospitable as possible to stop or slow the plants. Everywhere would be a parking lot. Roads when you could have them would be wide to block growth.