TL;DR: As of yet, there is no known relationship between the two women (beyond the shared surname), although it's possible both owe their name to Jeri Taylor.
Ensign Janeway (first name unknown) was a Sciences officer serving aboard the NCC-1701-D in the year 2369. We do not know anything else about her, save her appearance and the fact that her duties involved running a diagnostic on the sensor panels. This fact may indicate that she was in Astrometrics, although that is speculation.
In that same year, Kathryn Janeway (of USS Voyager fame) was already a Captain commanding the Oberth-class science vessel USS Bonestell, her first command - a fact which excluded her as Jean-Luc Picard's top choice to replace Commander Riker should he accept command of the USS Aries earlier that year.
As for special meaning, Jeri Taylor has gone on record1 as saying that the VOY captain was originally going to be named "Elizabeth Janeway" after the author of the same name, but could not be used for legal reasons and was changed. Several writers worked on TNG's "Man of the People" (despite not having credits)2, and Jeri Taylor was writing for TNG at that time, so it's entirely possible that she wrote Act III and simply used the name in both cases.
1: Star Trek: Voyager - A Vision of the Future
2: Captains' Logs: The Unauthorized Complete Trek Voyages
Now Let's Speculate!
That said, there ARE some striking similarities between the two, so let's have fun speculating about an in-universe relationship:
- Both serve in Starfleet
- Both share the same build, face shape, and hair color
- Both are Caucasian and have accents from the Midwestern US
- Both share an affinity for the Sciences Division
All of this is likely just a coincidence, but Captain Janeway does have a sister named Phoebe, who is four years younger. If we do some number-crunching, we know that Janeway left high school around 19 years before 2373 (when she gave up tennis). That means she was roughly 37 when Voyager made it back to the Alpha Quadrant. This would have put her about 27 in 2369, with her sister then being 23. Well, hey - what do you know? Ensign Janeway appears to be in her early-to-mid 20's in 2369. Could Ensign Janeway be Phoebe?
Note: Out-of-universe, there's no way the physical similarities are anything but a coincidence. The TNG episode "Man of the People" aired at least two years before Kate Mulgrew had even auditioned for the role of Kathryn Janeway.