Thank you very much, Mr. Zhang.

Differential difference equation is more general class than the delay differential equation. Differential difference equation may contain both the terms delay as well as advance terms simultaneously. e.g. y'+y(x-delta)=f(x) this is a delay differential equation (you can call it as differential difference equation also) where as

y'+y(x-delta)+y(x+eta)=g(x) is a differential difference equation but it is not a delay differential equation.

Dear Nicolas

I'm not going to model a particular system. Thank you very much for the help.

Delay Differential Equations is a apecial case of Functional Differential Equations. A differential Equation in which the differntial coefficient of dependent variable is the function of independent variable, dependent variable along with dpendent variable as function of delay parameter while in ODE is there is no delay term involved.

This lecture note may be helpful for you.

These notes may be helpful for you.