Both words are used - although "laundry" is more specific to clothes (and towels, bedsheets etc).
You use the washing machine to do the laundry. You can also "put a wash through the machine".
In context, you can "do the washing" - if the listener knows you're already talking about clothes. But if you're not already talking about clothes, then "do the washing" could refer to other things: for example doing the washing up (crockery and cutlery).
Perhaps the usage of "washing" is slightly more common in British English (and only within context, as above). However "laundry" is unambiguous on either side of the Atlantic.