I have bought a car in another state. And you must register in your home state/county and pay that sales tax. So no, you will not save on taxes. The states do this so they don't miss out on their sales tax revenue. You have to register in your home residence.
They can't do it easily on other smaller purchases, like clothing, etc. But it is very easy to do on a large purchase, like a car. And internet sales with no tax is another issue. Most people don't tell the truth on their state income tax returns which asks how much you purchased from a non-taxed internet sites. It is too difficult to police this, so people get away with it for now. But that will change soon.