Your trademark is the shorthand for your brand promised. It is often a registered trademark, but you have extensive intellectual property rights in a name without registering it formally with the patent and trademark office of your home country (ant others).
Your tradename, on the other hand, is the full legal name that you do business under. It it is the name your registered with the state or county in the USA, or with the government in other countries.
For example, Ford® is the registered trademark. Ford is the brand and you expect to find them at www.ford.com (though this is not always the case). But their actual company tradename is Ford Motor Car Corporation.
Similarly, Sun® is the trademark, but Sun Microsystems Inc. is their full tradename, and the name they are incorporated under in a given state.
Also, Starbucks® is the world’s most famous coffee shop name. It is their trademark as a name. It is also a trademark as a logo (the mermaid in a green circle). Starbucks Coffee is also a trademark of theirs. But the full tradename in the USA is Starbucks Corporation. In Japan, for example, their tradename is Starbucks Japan, but the brandname and the trademark are still Starbucks. They even have a lot of subsidiary corporations, and as a result the Starbucks trademarks are registered to Starbucks U.S. Brands LLC (of Nevada).
See also Trademark & Naming Resources for more in-depth discussions of more trademark related issues, as well as resources on where to file your trademark applications and company registrations.
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