You can always do either.
You can either recharacterize the contribution to a traditional IRA, or remove the excess (both by April 15). The consideration is what to do next.
If you recharacterize to a traditional IRA, it probably can't be deductible (if you are over the limit for a Roth, you are also probably over the limit for a deductible IRA). This gives you a non-deductible basis in your traditional IRA. If you have no other traditional IRA funds, you can then do a Roth conversion, which will not be taxable, and get the money right back in the Roth legally. (This is the so-called backdoor Roth IRA.) If you do have deductible money in a traditional IRA, it becomes more difficult to do a backdoor Roth, so you would probably just leave the non-deductible money in the IRA until you retire. Then, when you withdraw, part of your withdrawal would be non-taxable because it represents a return of the non-deductible contribution. This is tracked on form 8606 that you get with your tax return when you make a non-deductible contribution, and you basically need to keep copies of your 8606s for your entire life, they are an exception to the usual rule that you can discard tax papers after 3-6 years.
If you do a withdrawal of the excess, you could put the money into a regular investment account with a stock broker and let it grow for retirement. You would pay tax on the growth when you cash out, but you would not pay tax withdrawing the principal (basis). (I assume you still want to save for retirement, and not just spend it on goodies.)
If you can't do a backdoor Roth because you already have deductible IRA funds, there's really nothing wrong with doing the recharacterization and having non-deductible funds mixed in a traditional IRA. It still grows tax-free until you retire, you just have to remember to keep track of the extra bit of paperwork.
