Levee is almost always a noun, usually referring to an embankment used to prevent a river from overflowing. The word occasionally becomes a verb meaning to provide with or build a levee. There’s also a much rarer noun sense of levee used chiefly in British English: a formal reception.
Levy is usually a verb (inflected levied, levies, and levying), meaning (1) to impose or collect taxes, (2) to draft into military service, or (3) to declare and wage a war against. It also appears as a noun denoting things that are levied or the act of levying.
Examples
The Army Corp of Engineers is now working on building a levee system to prevent another hundred year flood. [The Atlantic]
Ryanair is to introduce a booking levy to fund expenses incurred by passengers due to delays. [Irish Times]
Angry flood-affected Queenslanders have slammed authorities and unregulated levee banks in their written submissions to the state’s flood inquiry. [Sydney Morning Herald]
The media companies have discussed levying a programming fee on Hulu just as they do with cable operators. [Vancouver Sun]
Long-term measures include building coastal levees to protect from tsunamis and installing watertight doors. [CNN]
San Francisco’s business tax is levied on payroll and covers salaries, wages, bonuses and other forms of compensation. [San Francisco Chronicle]