None whatsoever! 'Center' and 'centre' are exactly the same word, and the spelling has no effect on the pronunciation. 'Center' is used in the US and to a certain extent in Canada, while 'centre' is used in Britain and all other English-speaking countries. Other words which follow the same pattern are theater/theatre, fiber/fibre, for example.

The difference is based on what country the word is used in. Sometimes spellings or readings of different words change depending on what country the source material comes from. Center, will mostly be seen in the USA; Centre is seen in the UK and practically anywhere else in the world except for the USA. Here are other examples. Since the USA and the UK major English speaking countries of the world, I will use them as an example: USA: Color Program Math UK: Colour Programme Maths These are only a few examples, but there are more. Keep this in mind if you see words that look very similar to one another, but of course look them up in a dictionary first to confirm that meanings don't also differ as well.

Center is American, centre is British, I presume.

Same meaning but centre is correct in international and British English

Su.Ki is correct I'll add that in the U.S. one will ocasionally see a business using British spellings such as centre, or descriptions thought to be British such as 'Ye Old Pub, (the old pub) for example. The business owners do this to affect an 'Old World' sort of charm. Yes, it sounds silly but it's the truth!