There is no difference in meaning between center and centre. Center is the preferred spelling in American English, and centre is preferred in varieties of English from outside the U.S.
Some people do make distinctions between the words. For instance, some prefer to treat center as the word for a place or institution and centre as the word for the middle point of something. But while these preferences may be taught in some schools and are perhaps common among careful English speakers in Canada, the U.K., and elsewhere, they are not broadly borne out in 21st-century usage.
The following ngram graphs the use of center and centre in American books published from 1800 to 2008. It shows that center has been preferred for a century.
And this ngram showing the use of the words in British books during the same period suggests that center might be gaining ground in British publications.
American publications use center—for example:
The University of Southern Mississippi will announce plans Tuesday for a men’s and women’s golf training center. [USA Today]
He said his remark about his willingness to move the center, which was in answer to a question, was consistent with his previous statements. [New York Times]
Israeli and French filmmakers are making a comedy centered on the assassination of a Hamas operative in Dubai. [AP (dead link)]
And these publications from outside the U.S. use centre:
But the centre, who also missed the all-star game Sunday in Raleigh, N.C., at least took one big step closer to his return. [The Canadian Press]
Pfizer is to close its research centre in the UK with the loss of up to 2,400 jobs. [Financial Times]
Hickey could easily have started Cross at outside centre, but he is anxious to assess him as an inside centre. [The Australian]
The chief minister insisted that the Centre should verify these charges. [India Today]
The final phase of the Gautrain, which runs from Rosebank to Park Station and will in effect link Pretoria to the Johannesburg city centre, will open in the next few weeks. [Mail & Guardian]