I tend to think of a Test Bed as an environment configured for testing. It may be machines, operating systems, databases, etc - all configured as needed and ready to go for testing your application.
I tend to think of a Test Harness as something that specifically aids in your testing. For example, I might construct a framework that allows me to rapidly execute a series of test cases, each stored in an excel file. That framework might be considered a Test Harness.
Does that help?
(your mileage may vary, terms used by one shop may be different than in another shop)
Test Bed - an environment where System/ Regression/ Performance or any kind of Testing is carried out.
Test Harness - First place is a component in Test Bed. It is used as a anchor for testing a targeted component for testing. For example if you are familier with SOAP which is a xml based testing where a front end user cannot see the data. We can create or use a free ware which is available in market to get the xml input and then import to the custom front end to test the xml feed. This custom front is called as a Test harness. Test harness is mostly used in Top down and Bottom up testing techniques. Hope this helps.
I don't build Software but I make them work better, Testing is a Passion