Why do British and American English spell words differently?

Largely because of one man: Noah Webster, an American lexicographer who deliberately reformed American spelling in the early nineteenth century.

Before Webster, English spelling was inconsistent on both sides of the Atlantic. Samuel Johnson's dictionary of 1755 had begun standardising British usage, generally preserving the etymological spellings of French and Latin origins. Webster's American Dictionary of the English Language (1828) took the opposite view: spelling should be simpler, more regular, and closer to pronunciation. He also had an explicitly nationalist motive, arguing that a distinct American language suited a newly independent nation.

The changes he successfully promoted:

-our to -or — colour, honour, labour became color, honor, labor.

-re to -er — centre, theatre became center, theater.

-ise/-yse to -ize/-yze — though note that -ize is not actually American: Oxford has long preferred it in British English, and organize is perfectly correct in the UK. What is distinctively American is -yze in analyze.

Dropping doubled consonants before suffixes — travelled became traveled.

-ce to -se in some nouns — defence became defense.

Simplified digraphs — encyclopaedia to encyclopedia, oesophagus to esophagus.

Not all his proposals took. He also pushed for tung, wimmin and soop, which nobody adopted.

Some differences predate him or arose separately. Aluminium versus aluminum stems from a naming dispute involving the element's discoverer. Gaol and jail were competing British spellings.

Why the split persisted is that dictionaries and schooling on each side codified their own conventions, and mass literacy locked them in.

Practically: neither is more correct. Pick the convention your audience expects and apply it consistently — mixing them within one document is the only genuine error.