So you are asking about the advantages of making a non-deductible (i.e. after-tax) Traditional IRA contribution. There are two aspects.
First, as HartCO's answer mentioned, one advantage is that you can use it as an intermediate step in a "backdoor Roth IRA contribution", i.e. make a non-deductible contribution to Traditional IRA, and then immediately convert all of it to Roth IRA. This circumvents the income limit to make a direct contribution to Roth IRA, because both steps (non-deductible contribution to Traditional IRA, an conversion to Roth IRA) do not have income limits. Assuming you have no pre-tax funds in any Traditional/SIMPLE/SEP IRAs (and will not rollover any money into pre-tax funds for the rest of the year), this conversion will consist of all after-tax funds, and there will be no tax on the conversion. The end result will be basically the same as a regular Roth IRA contribution.
If, on the other hand, you leave the after-tax money in Traditional IRA long-term, then any gains will be considered pre-tax (i.e. the after-tax amount, the "basis", doesn't change as the value of the IRA changes). When you withdraw it, you will be taxed on the gains part as regular income. One difference with a regular taxable account is that any transactions within the IRA are not taxed. So whereas if you buy and sell stocks all the time in a taxable account, you would be taxed on each sale, and if you hold interest-bearing assets, you would be taxed on the interest every year, if you buy and sell or get dividend or interest in the IRA, you don't get taxed on it. It just goes into the value of the account and you are taxed on all the gains when you withdraw. So in some sense it is like an asset that you buy and hold all the way until retirement, getting taxed when you sell it at the end, except that if you were to hold an asset until retirement, a taxable account is better, because the gains are taxed as long-term capital gains and not regular income.