They both mean the same thing. "Enquire" is preferred in British English whilst "inquire" is used in American English.

As mentioned above, enquire and inquire are both verbs used to when talking about a request for information. In Australian English, "enquire" is far more common. However, when we make nouns out of "inquire" and "enquire", they are used in different contexts. For example, you can make an 'enquiry' to a restaurant by phone to ask if they have a table available for at a certain time. Or you can make an enquiry at the service desk at a supermarket to ask if they stock a certain item. It is an everyday request for information. However, an 'inquiry' is more like an official investigation, such as a government launching an inquiry into the handling of money, or an inquiry into the business practices of a large corporation.