The difference is stylistic & subjective. Both a vape pen and e-cigarette will vape, or atomize e-juice to vapor. A vape pen is a style similar to a kit pictured below.
E-cigarettes tend to be styled to appear more like king-sized tobacco / analog cigarettes. An image is below.
There is no objectively real difference between the two. Both atomize e-juice into vapor. Your choice of use comes down to your choice of style and your subjectivity. Others have come away from these two basic types of vaporizers and gone to mods.
Again, these conform to style & subjectivity, the basic use and functionalities being very much the same, if not exactly so. Vaporizers preform the task of heating e-juice until it atomizes and becomes vapor.
E-cigarettes, vape pens, mods are all considered vaporizers. When one vapes, one is using the vapor the devices produce. Vape is a verb and noun, one might call their vaping kit their vape, which they use to vape on.
Hopefully, this has helped somewhat. Links included are informational. I do not sell for, endorse any company. My linking back is simply courtesy for using the images. The images were chosen from a random search for generic terms.