A wealth tax is a tax on your net wealth.

Property tax is a tax on specific property (usually we are talking about real property, but I've paid e.g. property tax on a car). It is usually irrespective of wealth.

Someone who buys a house with cash and their neighbor who is underwater on their mortgage and at risk of insolvency will pay the same tax.

Note also that you rarely see property taxes on e.g. valuable jewelry because unlike real property and cars, both of which have ownership centrally registered, jewelry is not centrally registered.

If I understand you correctly, you’re saying the distinction on the “wealth” is specific to if the person owns the whole property. If they fully own the property then it could be considered a wealth tax. However, maybe people don’t since they have a mortgage so it’s not taxing them in their net wealth.

Right, that's one major difference.

Another major difference is that houses are both harder to hide and easier to value than many other forms of wealth.