While many of us get started with genealogy under the working assumption that surnames stay the same, more or less, we often find some point where we are stunned to discover that the last name we thought had 'always been the same' in fact has existed in the form we currently know it for only a few generations. There are a number of reasons that surnames change, often involving 'creative spelling' in bygone eras where spelling was not considered to be something unchanging, and creative spelling was not discouraged as it is today. In addition to that reason, some people only used spoken names and did not write their names, so the spelling could easily change. Every time cultures intersected, spellings could change, and immigrants frequently changed their names when they reached their new homeland.
So to answer your question about which spelling is correct, it's a matter of what era and what time you're looking at. The last name should be used as written at the time--provided that the spelling isn't a misinterpretation of someone reading a written record, and actually is the way the name was spelled at that place and time, for that ancestor.