In the prequel trilogy of Star Wars (the more recent three), the swordsmanship of the actors is great. There's lots of aerobatics and fast swordplay; generally impressive (albeit sometimes over-the-top) performances.
I disagree. The fight choreography was structured specifically to be flashy and full of special effects; there was very rarely any plot advancement or character development during any of the fight scenes (other than a character being killed or injured). I submit to you that the fighting is also highly unrealistic and you only haven't noticed it because of the aerobatics, fast and uninterrupted swordplay, camera cuts, etc. etc. distracting from how far the combatants come from striking each other.
Why is this? IMDB says the fight coordinator for the Phantom Menace was Andreas Petrides (IMDB film credits), who has a long and accomplished career as a Stunt Coordinator, with Nick Gillard as the Swordmaster. Wikipedia gives fuller credit to Nick Gillard for the sword battles. The key point is that neither one of them has real sword fighting experience.
Others have already pointed out the choreography for the original trilogy was by champion fencer Bob Anderson, who actually fenced on film in the Darth Vader costume in Episodes 5 and 6.
My biggest problem with the prequel's lightsaber duels come from the physics, as in this question. The blades would be very light, approaching weightless, and much faster than other movements of the body, and yet you routinely have lightsaber combatants performing all kind of compromising, exposing acrobatics in extremely tight proximity to others' blades without getting nicked. The excessive stunt work and jumping around makes the fights of the prequels even less realistic to my eye.
The duels in the original trilogy usually had the combatants facing each other, generally in sword-fighting stances (with the exception of one poorly executed spin move by Sir Alec Guinness). Fighters only talk when separated by safe distance (which is admittedly somewhat unrealistic, as a really interested combatant would open fire whenever you opened your mouth - but this is a storytelling conceit), then close with one another and fight with earnest lethality until separating. This is critically important to the realism of the fights in my opinion. When fighting in close quarters with weapons as swift and deadly as weightless blades of pure energy – Force enhanced muscles and perception notwithstanding - moving your weightless blade to block with a flick of the wrist is a far superior defense than jumping and spinning around.
On a different track, as I alluded to before, every duel in the original trilogy had a specific purpose to the plot and the characters in them. Luke flails around like an idiot at times because he is just a kid and an inferior fighter with less training and far less practice. Anakin began training before he was a teen and has been Darth Vader, Lord of the Sith for exactly as long as Luke has been alive.
Obi Wan was more than Darth Vader's equal in sword play, as evidenced by the fight at the end of Attack of the Clones, where Anakin learns his lesson excessive acrobatics by losing his legs. (Ironically, this move is incredibly similar to the video above, where Darth Maul misses the perfect opportunity to bisect Obi Wan in mid-air.)
In the re-match aboard the Death Star, Obi Wan's goal was not to defeat Darth Vader; he only needed to give Luke, Han and Chewbacca enough time to escape with Leia. Knowing that the tractor beam was disabled and being back in clear view of the Millennium Falcon, he knew his goal was accomplished when the others appeared with no Stormtroopers in between them and the ship. So he gives up, allowing himself to be slain knowing that he has a better chance of guiding Luke to defeat the Emperor than by slaying Vader himself.
Admittedly, the sequence is slower paced that the action sequences of the prequels, despite being minutes shorter than the shortest prequel duel; this fight features purpose and people actually trying to hit each other.
There is a similar purpose behind all the other duels in the original trilogy. (Another telling point is that each original film only had one major lightsaber duel per film, showing a restraint completely missing from the pre-quels.) I agree with the commentary from other answers on the original scene you linked. I would like to add that Luke's rage and the energy he gains from it are a great part of how he wears down and defeats Vader. Additional to the dark side Force boost Luke gains, his rage and pain have an effect on his father that penetrates Vader's defenses in a way his swordsmanship cannot, in my opinion, causing Vader to collapse on the walkway without even being struck.
I leave you with this quote from Wikipedia about the approach to swordplay in The Phantom Menace, which I feel was ironically captured by the original trilogy and sorely missed in all the prequels.
Stunt coordinator Nick Gillard was recruited to create a new Jedi fighting style for the prequel trilogy. Gillard likened the lightsaber battles to a chess game "with every move being a check." Because of their short-range weapons, Gillard theorized that the Jedi would have had to develop a fighting style that merged every swordfighting style, such as kendo and other kenjutsu styles, with other swinging techniques, such as tennis swings and tree-chopping. While training Liam Neeson and Ewan McGregor, Gillard would write a sequence to be an estimated 60 seconds long, meant to be among five to six sequences per fight. Lucas later referred to Jedi as being "negotiators", rather than high-casualty soldiers. The preference of hand-to-hand combat was intended to give a more spiritual and intellectual role to the Jedi.
The prequels have a much more stark view on good vs. evil, with the Sith being agents of pure evil in want of vanquishing by the Jedi. The redemption story line of Darth Vader in the original trilogy is much more nuanced, actually portraying some of the spiritual, negotiating and intellectual side to the Jedi. Also, the prequels show rather a large number of Jedi fighting and dying, exactly like the "high-casualty soldiers" Lucas claims they are not.
Gillards attributed theory of lightsaber combat combining "kendo and other kenjutsu styles, with other swinging techniques, such as tennis swings and tree-chopping" is rubbish. All of those are terrible ways to wield a light blade with unprecedented cutting power on all edges and points. Let's consider each in turn:
Kendo: Kendo is a stylized martial art descended from the katana and kenjutsu sharing many of the same problematic applications to lightsaber combat, which we'll consider next. But Kendo additionally limits the types of attacks and defenses that are allowed, which is rubbish for an actual fight.
Kendo techniques comprise both strikes and thrusts. Strikes are only made towards specified target areas (打突-部位 datotsu-bui) on the wrists, head, or body[...] Thrusts (突き tsuki) are only allowed to the throat.
Kenjutsu: "Kenjutsu (剣術) is the umbrella term for all (koryū) schools of the Japanese swordsmanship", which has been tailored and refined to the use and strengths of Japanese curved, single-edged swords (the most famous of which is the katana) over nearly 1,000 years. The debate over the strengths and weaknesses of a katana vs. other types of swords rages on; I'll not touch on it. However, I will say that a katana was very specialized towards cutting down opponents with lethal, single-swing strikes and preventing the blade from getting stuck in flesh, bone or armor. There the similarity to a lightsaber ends. Freed from real-world materials and physics, a lightsaber is a superior weapons and very different in the following aspects:
- Curved vs. straight blade.
- Single edge vs. omni-directional edge.
- Sharpness and strength. Katanas can be razor sharp, but they'll never be sharp and strong enough to plunge tip-first through a reinforced blast door (Episode 1, trade ship bridge incursion attempt by Qui-gon and Obi-wan).
- Blade weight and balance; katana typically weighs over a kilogram, balanced over the length of the blade. Lightsaber weight and balance is not canonically established, but it's very difficult to imagine a plasma/energy blade has similar properties to solid steel.
- Blade length. Variable-length lightsabers are established in canon. Any fixed-length blade will be a very different weapon to handle than a variable-length blade.
Any one of these would dictate noticeable differences in fighting style. Considered all together, Jedi and Japanese swordplay should be completely incomparable.
Tennis: really? I guess that explains why the combatants in the prequels are frequently fighting so far apart.
Tree-chopping: Jedi use light sabres not lightaxes. Furthermore, tree-chopping is best done with heavy strikes with as much mass and force behind them. You wouldn't swing a chainsaw at a tree like an axe; you wouldn't even need to swing a lightsaber to cut a tree down, just touch it. It takes less than a pound of pressure to cut skin with a conventional steel sword; how much pressure do you think it takes to cut skin with a lightsaber?
"other swinging techniques": Hopefully I've made it clear that speed of attack is a critical advantage of lighter weapons. A thrust is a faster attack than a swing, since it moves in a straight line rather than a curved arc. That is why foil fencing (lightweight blade restricted to stabbing with the tip) or sabre fencing (similarly lightweight blade, but allowed to score hits with either edge) would be the modern day martial art with the most similarities to lightsaber fighting.
In conclusion, the major differences you observe between the prequel and original trilogy are by design. The original trilogy uses lightsaber as a storytelling vessel, exploring the meaning of the Force and the character development of Father and Son Skywalker. The choreography is minimal and pointed, the realism of the foil fencing technique taking a back seat to the fighters themselves. The prequel trilogy features Hollywood/Kung Fu choreography as action set pieces, designed to show off the acrobatics and special effects, largely superfluous to the story line. In short, the fights are so different because they are different styles of movie.