So what is the difference between a land tax and property tax? After all I looked at my tax assessment recently and there is a value for the land and a value for the house itself?
This might seem like a minor difference but it actually isn't, because taxes generally cause changes in behaviour. Sometimes those changes are desirable: Carbon taxes, "sin" taxes, etc. More often the change is undesirable (as when we tax investment, wages, or consumption). (And, very frequently, those changes in behaviour results in the actual burden of the tax shifting. Payroll taxes are levied on businesses, but nobody actually thinks they pay them.)
Tax housing, and you'll get less housing. Since people like housing (that's why they build it), that's considered a Bad Thing. Tax land, and you'll...get the same amount of land. With a few very minor exceptions, the amount of land is fixed. No distortions, no deadweight costs, and no real ability to shift the economic burden of the tax.
As a home owner I want a garden, but I don't want to pay for it so I build it on top of my house and make a basement to put my car in - I now shifted something like 8/10 of my tax burden because I use less land.
Now, I don't believe that this is sufficient as an alternative to all other taxes, but it does have many desirable properties, and your suggested behavioural changes are amongst them: Optimizing usage of land.
So in effect I have shifted most of my tax-burden off to others, land will have very little value (because it will have to be taxed more and more heavily).
Of course you can mix it with older taxes, but then you have the problems you had with them too.