Process time delay is where the process is still producing, but at a reduced rate.
Process dead time is the process is completely down, no production.
Both a perspectives and definitions may vary slightly for each situation applied.

For example in a manufacturing facility, the entire plant or just one line may go completely down ('dead time') several times a day. But the industrial engineer may report if as 'process delay', because the product made it out the door eventually, just delayed. Companies should set their own specific definition so communication and transparency are improved.

Yet when talking process automation control, the two phrases take on a whole new meaning...
"Dead time is the delay from when a controller output (CO) signal is issued until when the measured process variable (PV) first begins to respond. The presence of dead time, Ó¨p, is never a good thing in a control loop." Ref:#-Link-Snipped-#

Would be like the PID loop in your car's cruise control. When you start up a hill, the cars instead of maintaining a constant speed via the PID control loop, instead it slows down a couple miles per hour, then guns it to catch up. That would be a 'dead time' in your cruise control's PID loop.

With the PID control scenario, variations in the process time delay are the 'time delay' being referred to.