The differences are mostly semantic in nature.

Heathenry is a reference to Heathenism, which is an umbrella term for various spiritual movements focused on revitalizing different pre-Christian Northern European practices, such as Slavic Heathenry, Baltic Heathenry, Teutonic (Germanic) Heathenry, Anglo-Saxon Heathenry, and Norse Heathenry.

Norse paganism is another word for Norse Heathenry. It’s also sometimes called Norse polytheism. Some people make distinctions between all three of these terms, but these distinctions are mostly arbitrary.

Ásatrú is the official name for Norse paganism, similar to saying “Christianity” or “Buddhism.”