Who In Florida is responsible for paying for damages to the vehicle one has insurance the other does not?

If the insured vehicle had collision coverage, the most direct route by which to have the damages taken care of would be by making a "first-party" claim against your own insurer. If you did that, you would be responsible for the payment of the deductible. When the insurer pays for the repair of your car, it acquires the right to pursue the wrongdoer to get its money back; that process is called "subrogation". More often than not, the insurer also attempts to recover your deductible.

Collision coverage is not usually mandatory unless the car is financed. In that case, the lienholder probably will require it if only to protect the value of the collateral on which it has loaned money.

If you had no collision coverage, you may file a lawsuit against the owner/driver of the at-fault vehicle. If the driver was using the vehicle with the owner's express or implied permission (assuming that they are different people), both parties are potentially liable and you would be able to seek recovery from both. That is, although you could only recover your damages once, recovery could be had from the driver, the owner, or some combination (called "joint and several liability").

The suit would be to recover the cost of repair and other damages that may have been incurred (such as reasonable rental expenses, loss of use (if there was no replacement car rented), towing charges, etc). The court in which suit would be filed depends upon the amount of damages. It is always a good idea to see if the other party is willing to voluntarily settle the claim, as it never a certainly how a lawsuit will turn out, or whether, even if you win, you will be able to collect.

If you have to file a lawsuit, it would likely be based upon a legal theory called "negligence", which essentially means "carelessness". In Florida, which adheres to the theory of "comparative negligence", your damages will be reduced depending upon how much negligence is attributed to you (if any). Therefore, your realistic assessment of the degree of negligence of you and the other driver will be among the factors in determining how hard you press the claim and when or if you compromise.