I would say that "je m'assieds" is now more commonly used, but both forms are perfectly correct and understood everywhere in France. French schoolchildren were also confused as to what form they had to pick, and were answered "just pick the one your Mum uses at home".
The reason is that there was a linguistic variation between northern and southern French (that doesn't exist any more except for the regional French accents). The north spoke a variety known as langue d'oïl, the south spoke the langue d'oc (after the ancient forms of 'oui' meaning yes).
The variations in conjugation for the French verb "s'assoir" exist due to historical reasons and regional differences in the language. While "je m'assieds" is the most commonly used form, "je m'assois" may be used in certain dialects or by individual speakers. Both forms are correct and acceptable in modern French.