Welcome OP! There are two great posts (here and here) from years past that dive down into some of the differences. We are named after your city and were supposed to be the "Manchester of America" based on your industrial past.

We're a much smaller city of around 115,000 people. Similar to your Manchester, our city was based around textile manufacturing for many decades. Our location was decided because of a significant drop in elevation in our river (Merrimack River) that allowed for a canal system to be constructed parallel to the river to move maritime boats up and down the river without going against the current. This significant current was later used to power textile mills. We were a company town run by the Amoskeag Manufacturing Company in the early days, with most of the city's design planned before any major construction took place (our grid system of roads and architecture can be attributed to this as well). After the original company went bankrupt in the 1930s, Manchester switched to wartime production during World War 2. Most of the manufacturing in the city was gone by the 1980s and the Millyard as we call it started to make the switch from industrial to offices/apartments, where high-tech companies are now based such as DEKA Research (engineering firm that helped develop the Segway through a subsidiary), Dyn (software company acquired by Oracle) and ARMI (tissue regeneration).

Our city isn't well known outside of the northeast US, but it serves as an important regional city in New England and is the largest center of commerce and business in Northern New England. We generally say New Hampshire since we are a small state and no one town or city is well-known nationally that I can think of.