On average across the year, no, New Hampshire is not hotter than Nebraska . New Hampshire has an average temperature of 9°C/48°F and Nebraska has an average temperature of 11°C/52°F.
New Hampshire's hottest month is July, with an average maximum temperature of 29°C/84°F, which is not hotter than Nebraska's hottest month (also July, with an average maximum temperature of 31°C/88°F).
On average across the year, yes, New Hampshire is colder than Nebraska . New Hampshire has an average minimum temperature of 3°C/37°F and Nebraska has an average minimum temperature of 4°C/39°F.
New Hampshire's coldest month is January, with an average minimum temperature of -9°C/16°F, which is approximately the same temperature as Nebraska's coldest month (also January, with an average minimum temperature of -9°C/16°F).
On average across the year, yes, New Hampshire has more rain than Nebraska. New Hampshire has an average annual rainfall of 1196mm and Nebraska has an average annual rainfall of 764mm.
New Hampshire's wettest month is October, with an average monthly rainfall of 130mm, which is drier than Nebraska's wettest month (May, with an average monthly rainfall of 133mm).