Benzene is carcinogenic and was removed from the consumer market 40 years ago. Before then it was often used as a paint and varnish remover.
Benzine is another name for naphtha in the US, though the term is rarely used in this manner anymore. It’s also a common name for gasoline in England.
Naphtha (benzine) is a faster evaporating, less oily (“drier”) form of mineral spirits. It’s not dangerous if used in moderation.
Here’s an easy way to remember which is which. Benzene is spelled with an “e” as in dead. Benzine is spelled with an “i” as in alive.