They are the same word and are used interchangeably. In the US, the word is usually spelled center unless it is in the name of a location or street. When it is in the name of a location, it is usually spelled that way to give it a fancy sounding flair or element of style. Little irregularities like this are a big part of what makes English a difficult language to master.

They're alternate spellings of exactly the same word, so it makes sense they'd have the same pronunciation, no? Sometimes just the pronunciation varies between dialects, and sometimes just the written form, in any language. There is a long list of words that end in -ter in American and -tre in British. Lustre, spectre, etc.

Is it a rule that these words (the tre, vs. ter words) have the same pronounciation? Yes, they do.