Wikipedia page on deep space exploration summarizes it pretty well:
Deep Space Exploration is the term used for the exploration of deep
space and which is usually described as being quite distant away from
Earth, within or away from the solar system.
It is the branch of astronomy and space technology which is involved
with exploring the distant regions of outer space. Physical
exploration of space is conducted both by human spaceflights and by
At present the furthest space probe mankind has constructed and
launched from Earth is Voyager 1...
Staying with broader definitions, and distinctive to the definition of deep space, Wikipedia page on interstellar travel says the following:
Interstellar space travel is manned or unmanned travel between stars.
The concept of interstellar travel via starships is a staple of
science fiction. Interstellar travel is conceptually much more
difficult than interplanetary travel. The distance between the planets
in the Solar System is typically measured in standard astronomical
units, while between the stars is hundreds of thousands of ㍳ and often
expressed in light years. Intergalactic travel, or travel between
different galaxies, would be even more difficult.
If we analyze current questions tagged deep-space or interstellar-travel, we can observe them following similar patterns:
Questions about deep-space are mostly related to problems of space travel and observations that are in one way or another related to the distance between the object in question and the Earth, for example communications delay, radial resolution, e.t.c.
Questions about interstellar-travel mostly inquire about possible methods of propulsion in the interstellar medium, methods of movement through timespace that possibly cannot be used in areas of significant gravitational pull or other influences of a star and its orbiting bodies, reaching escape velocity to leave an area of a star (granted, mostly our Solar system), e.t.c.
I.e., the difference is deep-space denotes a distance relative to a point of origin or observations (in our case the Earth and its near space), whereas interstellar-travel the distance from any star, star system, galaxy, or any other mass body and its influence on another.
I think the tag should stay, and is currently also used (for the most part) to respect this differentiation described here.