The question is deceptively simple, deserving a thorough answer. Read on...
Yamato 2199 is neither a slavish remake of the original, nor a "reboot" or "re-imagining" in the vein of typical TV and cinematic returns of older shows. Instead, it might be best described as a careful, even loving, reconstruction of the series for the modern era. Even the music is composed by Akira Miyagawa, the son of the original composer Hiroshi Miyagawa -- that gives an idea of the sense of continuity embraced by the series. (And, interestingly, the younger Miyagawa recreated the scores note-for-note by ear from recordings because the original sheet music has been lost.)
I'll try to avoid any significant spoilers...
The familiar characters are all present, with many new ones added and minor characters re-defined and in many cases fleshed-out in much greater detail.
There is really no significant gender-bending. One fighter pilot character who was basically mashed-up out of two different ones during the original two series due to haphazard editing and re-writes from the TV series to the movies, (though in different ways in Japan and in the American re-release) has been retconned into two different individuals, one of them female. That's about the extent of it.
There is also a retcon/rework among the Garmillan/Gamilas/Gamilon cast of Talan/Masterson into two individual, related characters which eliminates the occasional who's-who debate there.
There is a subtle re-work of the characterizations of Kodai (Wildstar) and Shima (Venture) that balances their natures more. Shima is more aggressive, Kodai more introspective from the start. It allows both characters to play out a wider range of motivations and reactions, altogether for the better -- they are multi-dimensional characters now, less cartoonish.
Yuki (Nova) is much more complicated of a character than before. The romance that blossoms between her and Kodai gets off to a far rockier start, but the characters' discoveries of each other also play out to greater effect than a simple boy-gets-girl tale.
Mamoru Kodai's (Alex Wildstar's) martyr role is still there, still a deep emotional motivation for his brother, and is taken several steps beyond into a greater significance as well.
Captain Okita (Avatar) is still the steadfast, introspective ship's captain, still motivated by deep sadness and a sense of loss tinged with unfailing hope and determination. His bold tactical skill plays out even better than before; he is authentically the ship's "Old Man" captain in every way.
Interestingly, a few characters from the Comet Empire story arc make an appearance here. Should the new series progress, there are a few anchors for it placed already.
Entirely new characters, many female, have been added on both sides of the conflict to add depth and to dig out of the male-dominated rut of the original shonen anime style of the original series. There are also an additional Iscandaran, female as all others have been -- Isacandar is still a dying race.
The Garmillans (as the latest translation goes) are treated to much greater detail. They're not the cardboard-cutout bad guys save for Dessler/Desslok as before. They still maintain their WWII-era Nazi overtones, though it's rendered in a richer, multidimensional style. There are reasons behind their motives, political and otherwise. In fact, the Garmillan subplots delve into internal power struggles, questions of honor and of right and wrong, and conflicts that drive the main story. Dessler's pride and hubris are explored more deeply -- and we even see (extremely) fleeting moments of doubt an self-examination in him -- making him every bit as intriguing as ever, and with new insight. Other Garmillans become interesting characters in their own right. There are still a few fawning lackeys and boastful idiots among them, but they server the plot well.
This time around, the Earth forces aren't 100% in the right. Earth's political factions exist, and have contributed to the Garmillan conflict. There are shades of gray this time, even among the Yamato's crew, which add some very good subplots of self-examination and redemption.
Over-arching themes abound, and are more complex. The dualities of right/wrong, oppression/resistance, twin worlds Garmillas/Iscandar, even Yuki's (Nova's) mistaken identity for a certain Iscandaran are woven through the story in ways that add depth and introspection that was barely hinted at in the original series.
There is also a deep, underlying and very Japanese theme of the question of the use of great military power. The spectre of WWII runs deep in the new series, and is examined thoughtfully. The dates of the Yamato's launch and return revolve around the date of the Pearl Harbor anniversary. The Yamato launches, secretly armed with an adaptation of peaceful Iscandaran technology (the Wave Motion engine) turned into the most powerful weapon in the galaxy -- the Wave Motion Gun. How the crew of the Yamato use that power leads to both ruthless self-examination by the Captain and crew, and by Starsha of Iscandar. The honor -- or lack thereof -- versus necessity and bravery of covert attacks by both Earth and Garmillas are replayed and examined with much soul-searching by characters on both sides. Combatants from both sides meet, sometimes unexpectedly, and must come to terms with their own motivations, and even similarities. These scenes play out in an amazingly unblinking philosophical tone.
There is a clear attempt to ground the story in better science than before, without diminishing the original concepts. In some cases, the early theories that the original played on have simply progressed, and the new story takes the advantage. There is a cleaner presentation of how the new Yamato is truly a new ship built inside the hulk of the old -- though throughout we see details of the old ship retained (such as a WWII-era builder's plate on the interior armor skirt of a gun turret) that have been retained, lending a realistic sense of history and shipbuilders' tradition.
Instances of deus ex machina that made for pivotal plot points in the original are carefully played out with all the sense of wonder as before -- but -- by the end, they all have an explanation, scientific and spiritual. The way these critical scenes are handled is often startlingly philosophical and cinematically masterful. The story telling technique banishes any underlying weakness in the original plots and instead injects a sense of wonder that re-captures the impact of the original for a modern audience. This alone makes the series well worth watching.
Certain pivotal scenes play out nearly exactly as the originals or are improved upon, if it's possible. For instance, the death of Captain Okita (Avatar) is scripted and timed virtually frame-for-frame to the original -- the impact simply can't be improved upon. Others, such as the Yamato's blast-off from Earth are both tightened and given added detail. The Rainbow Galaxy battle is fleshed out with even greater detail of the Garmillan carrier plane launches -- depicting them intentionally with all the details of a real deck launch, with deck crew signals, catapult operations, and signal lights for the pilots -- all to drive home how the "enemy" isn't so different from "us".
If there is any complaint to be made at all, it's probably that the gratuitous fanservice could have been cut back. But -- the originals weren't without it, and there is a sort of tradition for it to stand on. And in some instances, it's used to good effect to break up dramatic tension where any other form of comic relief would have come off weak or forced. In other words, humans will be humans... despite life-or-death circumstances. So, even the fanservice manages to nudge its way into the duality thread that runs so deeply through the story. I'll give it a pass this time...
All in all, Yamato 2199 succeeds brilliantly as a modern update that does not diminish the original, yet stands on its own as a definitive interpretation. This is truly a worthy Space Battleship Yamato series that earns it's right to carry the name. In fact, it goes so far and does so well that it virtually defines how a remake should be done with respect to any movie or series, anime or otherwise. It is that good, startlingly so, even.