They are relatives. Garden persicarias can be attractive space fillers and some have good leaf markings. Some are small ground cover plants ad others are much taller and gangly. Some can be invasive when happy but won't push through tarmac, concrete and buildings like JKW. There is also a weed form which I have in plnety here but haven't had in other gardens.
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There's more info on JKN here form the RHS - https://www.rhs.org.uk/advice/profile?pid=218
and here - https://www.rhs.org.uk/science/conservation-biodiversity/invasive-plants
Persicaria comes in many forms - https://www.rhs.org.uk/Plants/Search-Results?s=desc(plant_merged)&query=persicaria&form-mode=true&context=b%3D0%26hf%3D10%26l%3Den%26q%3Dpersicaria%26s%3Ddesc%2528plant_merged%2529%26sl%3Dplants&page=1
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