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From Ohio Tax Dept website:
A use tax is similar to a sales tax in its application. Use tax is imposed on the "storage, use or other consumption" of all tangible personal property and the receipt of certain services that are subject to sales tax. Purchases that are properly exempt from sales tax are also exempt from the use tax. Use tax must be paid on all purchases made by Ohio residents and businesses if the proper amount of sales tax has not been paid to the vendor, seller, or service provider. The use tax rate is equal to the sales tax rate in effect in the county where the property is used or benefit of the service is received by the purchaser.
The most common situation that gives rise to a use tax liability is when an Ohio customer makes a mail order or internet purchase from an out-of-state seller. Use tax is also due when the purchaser improperly claims exemption from the sales tax or if the sales tax paid is less than the total use tax in effect in the county where the item is used or benefit of the service is received.
One of the main reasons for the use tax is to protect Ohio vendors from unfair competition from out-of-state sellers, since the in-state merchant is required to collect sales tax when selling to an Ohio resident or business. All states that have a sales tax have a use tax.
6. How is the use tax paid?
Many out-of-state sellers are registered with the State of Ohio for the collection and remittance of the use tax. However, if the proper amount of sales or use tax is not collected at the time of purchase, the amount of tax due can be paid directly by the consumer to the State of Ohio. A consumer who makes regular purchases upon which the proper amount of sales or use tax is not collected by the seller should apply for a consumer's use tax account with the Department of Taxation and begin filing use tax returns. If the purchases are isolated, the consumer should make a voluntary payment of the tax by sending a payment of the amount due to the Ohio Department of Taxation along with a description of the purchases made and the date of purchase. Use tax on purchases can also be paid on the Ohio income tax return. A voluntary payment form (VP USE) is available on our Web site at Tax Forms.