Would a stargazer on mars see the same constellations?

Yes. It's only a tiny distance on the scale of stellar distances so you would notice no difference. You certainly don't when the Earth is on the opposite side of the Sun from where it was six months before, and that's nearly 200 million miles different.

It has been worked out that even if you went 4 light years away to the nearest star, Proxima Centauri, all you would notice is that Centaurus no longer contains Alpha Centauri (because you're right up close to it!) and Cassiopeia would contain an extra star - the Sun. Maybe some other nearby stars would appear to have shifted a bit, such as Sirius, but that's about all.

For about half the year yes, the other half no.

But while close, parallax or binocular vision is what allows us to gauge depth perception, the reason we have two eyes. The background from one eyeball to the other is just different enough from the other against a background we can tell the distance. Whether you're looking at a star,a planet or a turkey leg on a plate, this constantly shifting perspective is an important tool to astronomers to judge distance.

For a good part of the year the night skies are COMPLETELY DIFFERENT. When Earth and Mars are on opposite sides of the sun, their nighttime skies are exactly the opposite. Were you to show me a night sky on Mars today, i could tell you with reasonable precision then where in its orbit in relation to earth it is. And deduce is it close to Earth in its orbit or far away as they rotate around the Sun.

good question

EDIT:

That tiny difference is enough for Astronomers, once we knew with accuracy the size of earth orbit (an AU), to measure distances using the pythagorean theorem. Proxima Centauri has a parallax of 0.762 arcsec, and therefore you can easily find its 1.31 parsecs away.

Some other examples of distance measurement by the parallax method are 61 Cygni at 1/3 of an arcsec, distance 3 parsecs, Barnard's Star at 1.8 parsecs = 5.9 light years. Barnard's Star also exhibits a large proper motion.

The distance at which parallax can be reliably measured has now been greatly extended by space-based instruments like the Hipparcos satellite. Or if you wanted to sue Mars to build the right triangles. The reality is constellations are different from Mars.

While a novice might say its of no significance, a scientifically minded individual would look and say what can I learn about the universe from this data.

Someone on Mars would see exactly the same thing as someone on Earth. A distance of 250 millions miles is trivial on an interstellar scale. For example, the nearest star is about 25 trillion miles away. If you do the division:

250,000,000 miles / 25,000,000,000,000 = 0.00001

That means that the largest Earth-Mars separation is only about one hundred thousandth (1/100,000) of the distance between Earth and the nearest star! In fact, if you could travel to Pluto (six billion millions) the view would still be the same.

Yes of course they would. We see the same constellations as we move around the Sun.

If you were standing on the beach looking at a ship you could just barely see out on the horizon, then leaned your head just 1 millimeter closer to the ship, would it look any different? Looking at stars light years away, even if you went all the way out to Pluto, would be a MUCH, MUCH smaller difference than leaning your head out a millimeter to look at that ship.........

Yes - The stars are at effectively infinity compared to distances in the solar system.