Both economics and finance involve money. The difference is that economics (conventionally treated as a singular noun despite resembling a plural) is large-scale or abstract, while finance is particular, usually involving the monetary matters of a person, company, government, or household. For instance, a profitable business has financial success, and a country full of profitable businesses has economic success.
Examples
In the First World, including the United States, Japan, and the Czech Republic, economic growth typically proceeds at a steady, sustained pace over decades. [NPR]
Couples who seem to be successful at marriage are usually good with their finances. [Winnipeg Free Press]
The basic laws of economics are threatening to pull the eurozone apart, just as politicians are trying to pull it together. [BBC News]
A good set of financial results from BSkyB hardly counts as news these days. [Guardian]
The economics of high-speed rail are determined by a simple formula: the number of people carried by distance travelled. [Sydney Morning Herald]
Worries about inflation, the slumping dollar and the U.S. government’s finances have sent the price of silver soaring . [Los Angeles Times]