Why there are two different lists at all
Birthstone customs existed in fragmented, regional form for centuries before anyone standardised them. In 1912, the National Association of Jewelers — the organisation that became today's Jewelers of America — published one official list, intended to settle the disagreements between the various traditions that had grown up separately across Europe and America. That 1912 list is what is now called the "modern" list, and it is the one printed on most jewellery-store charts and used in most retail marketing.
The "traditional" list is the older usage the 1912 standardisation was reacting to. It was never a single unified system either — it is really a snapshot of the most common Western custom before 1912 — but it is still published alongside the modern list because the two genuinely disagree for six of the twelve months, and neither version has fully displaced the other in circulation.
Where the two lists actually differ
Six months carry the same stone on both lists: January (garnet), February (amethyst), April (diamond), May (emerald), July (ruby) and September (sapphire) have never been contested. The other six are where the lists genuinely diverge. March is aquamarine in the modern list and bloodstone in the traditional one. August is peridot (plus spinel, added later) against sardonyx. October is opal and tourmaline against opal alone. November is topaz and citrine against topaz alone — the closest of the divergent months, since topaz appears on both. June carries pearl on both lists, but the modern version adds moonstone and alexandrite as alternatives. December is turquoise on both, with the modern list adding zircon and, since 2002, tanzanite.
The trade bodies that maintain the modern list have amended it twice in living memory, both times by adding an option rather than replacing an existing stone. Tanzanite was added as a December birthstone in 2002, decades after the gem was first identified near Mount Kilimanjaro in the late 1960s and needed time to become widely available. Spinel was added as a second August stone in 2016, partly because good spinel had become more available and partly because it had long been mistaken for ruby and sapphire in older jewellery, then reclassified once mineralogy caught up with what was actually in old pieces.
What a birthstone list is, and is not
None of this is graded, priced, or backed by any gemological property — a birthstone list is a naming convention, not a statement about a stone's hardness, rarity or value. A December baby's "birthstone" is turquoise, zircon or tanzanite depending purely on which list is consulted, and none of the three is more correct than another; they simply come from different points in the same custom's history.