There is likely no difference between "experimental design" and "design of experiments" except perhaps "design of experiments" applies more theoretically and generally to all experiments while "experimental design" usually refers to the design of one particular experiment. However there is no substantive difference between the two, and you could use the two interchangeably to refer to the same thing.
Experimental design is, ultimately, about how your experiment is constructed. If you want to test the effects of a drug, you may split people into two groups and administer the drug to one of the groups. You may decide instead to have three groups, where one group gets the drug under investigation, one group gets a drug that is already in the marketplace, and one group gets nothing. This would be a difference in "experimental design" because you are deciding how many groups you want to include.
When splitting people into groups, you may consider certain variables (like age or gender) for which you wish to control, so that means you might make sure that each group is 50% men and 50% women. This is a design choice - do you want to control for age/gender/other variables?
Sample size is also an aspect of experimental design. If you split people up into groups, should the groups be equally sized? Is there a reason, due to cost or availability of the drug, that your group reviving the drug can only be so big, so the group receiving the drug and the group not receiving the drug may be of different sizes?
I've briefly mentioned a few basic aspects of experimental design here but entire areas of study are devoted to the topic, so sample size, number of groups, and whether or not to control for certain variables are certainly not an exhaustive list. What I recommend is to answer the question "What is my goal here?" Depending on your goal (i.e. see if a drug has a particular effect on people), I would use that as the starting point and develop the simplest experimental design that allows you to completely answer your question.