the Turbo Tax form for 1099-R doesn't match the numbers on the issued 1099-R (after box 11)
(Boxes 12-17 on 1099-R are different from boxes 12-19 on Turbo Tax form)
Box 14 on the 1099-R is titled "State taxable amount"
Box 16 on Turbo Tax says "State distribution amount"
are these supposed to be the same thing?
Yes, in most cases this would be the same thing. There are few States that it would show different information in this. If you have any questions that are related to your particular State and if they treat for whatever reason Total Distribution and State Taxable Amount differently, you can go to that State Dept. of Revenue site and should be able to get full clarification on this (as it will also depend on other information contained on the 1099-R that I am not aware of). But for the tax system, yes.
Make sure your 1099-R is not a CSA-, or CSF-1099-R. IF it is, then you are entering your in the wrong form type in the software.
But if its just a standard "2020" 1099-R, the box numbers 12,13,14 (2019 forms) were moved to 14,15,16 on the official redesigned IRS approved 1099-R.
A few issuers are sending out 1099-R forms using the old box number designations...failing to upgrade to the proper 2020 format.....so you would put your box 12,13, 14 values into the software's 14,15,16 (though 16 is empty on many 1099-R forms...where the state automatically uses box 1 or 2a instead).