There are two kinds of taxes you can possibly avoid buying in duty free shop:
- Excise tax — this typically applies to alcohol and tobacco, in some cases perfumes (also fuel and firearms, but these aren't sold in airports ;-)
- VAT (sales tax) — if and only if you're traveling from EU to destination out of EU. It's also possible to get VAT refund for stuff that you bought outside of airport, but there are restrictions.
Excise taxes vary widely, but generally they are quite high on tobacco, for example US Federal Excise Tax is $2.11 per pack of cigarettes. And there can be state excise tax on top of that. With alcohol the excise tax is calculated based on alcohol content, thus it's higher for distilled spirits, and practically irrelevant for wine of beer. And still, even for distilled spirits it's much less in relation to total price than in case of tobacco. Eg. US Federal Excise Tax for 750ml bottle of 80 proof spirit is $2.14.
Standard VAT ranges anywhere from 15% to 25% depending on a country.
No for "is it cheaper" part: from my experience it's almost always cheaper in case of cigarettes, even if you don't get VAT return. In case of alcohol if you don't get VAT return, than usually you can get better price at a supermarket, than the difference made by not paying excise tax. In case of not paying VAT, YMMV.
In case of all other products: even if you're getting VAT return, still the price in airport shops is that much higher, that you're not gaining anything at all. It's rather kind of last resort for these who either don't have time or simply forgot to shop for gifts.