In terms of meaning there is no real difference, they both translate as 'shrine'.
However, Jinja is just a general term for any shrine, whereas Jingu denotes a more prestigious establishment, usually linked to the Imperial Household (Japan's equivalent of a Royal Family) or enshrining an Emperor.
There are lots of other words for shrine: Yashiro, Miya, Hokura, Taisha... all indicating slight differences in function and status.
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