When an extraneous factor influences an experimental group versus a control or comparison group differentially, it means that the influence is different for the two groups.

For example, let's say that your experimental group is a group of alcoholics who have voluntarily admitted themselves to a treatment center. This particular treatment center allows no radio or television, nor any other contact with the outside world. Music is allowed in the form of tapes, records, CDs, etc., but these are screened by the treatment staff. Magazines and newspapers are likewise prohibited.

You were unable to randomly assign persons within the sequestered center to experimental and control groups, and you were also unable to find a true control group setting that was identical to the experimental group, so you had to settle for a comparison group in a treatment center that does not exercise control over television, radio, magazines, etc.

During your experiment (just what does not matter for this discussion), JAMA prints a report of a drug that reportedly "cures" alcoholism, and the story is promptly reported by news magazines and television.

Do you think your comparison group is likely to be affected by this event differently than your experimental group?

This is an example of a differential effect of the extraneous factor history. If your experimental group setting and control group setting had been identical, or you had been able to randomly assign persons within either treatment center to experimental or control group, the differential effect of this news (history) would not have occurred.