Per Merriam Webster:
Town House (aka Town home): a house that has two or three levels and that is attached to a similar house by a shared wall
Row house:a house in a row of houses that shares a wall with the houses next to it.
Per
Portland Oregon's Bureau of Development Services (see page 10 and 11): A Townhouse is a single family unit constructed in a row of attached units on a single tax lot. A Rowhouse is a single family unit constructed in a row of attached units separated by property lines. However, your municipality may have different definitions.
So it appears that physical descriptions of the construction of the home is the same. However, a municipality may have defined them differently for tax purposes. Therefore, you will need to check how or if your municipality has different definitions.
How does it apply to anything? Depending on your municipalities definition, it may affect how they are taxed by the city. I don't think it makes a difference to the Association. Perhaps this is the way the individual likes to demonstrate their knowledge (like the character Sheldon on the Big Bang Theory).