Where to begin?
Let's start with the temporal problems:
In the OT, it is implied that Leia knew her mother for several years before her death, whereas in the PT Padme dies moments after she was born.
The Old Republic existed for over one thousand generations in the OT, whereas in the PT it existed for only a thousand years.
Owen, Beru, Anakin, and Obi-wan age forty years in the twenty years between trilogies.
Character Problems:
It is implied that Obi-wan trained Anakin because of hubris (and in other versions of Jedi it confirms this), and not merely to fulfill a promise to his dying master (who, by the way, was never mentioned in the OT).
Obi-wan and Yoda simply run away at the end of Episode 3, even though they were prepared to kill both Anakin and the Emperor directly prior to this. They even meet up after their injury-less defeats, and they never consider returning together to destroy the Emperor while Anakin is incapacitated. Instead, they wait for twenty years for Luke to come of age (which actually would be considered too old to begin Jedi training anyway), and never even consider training Leia. Obi-wan even considers Luke their "only hope", even though the PT confirmed the existence of female Jedi, making Leia a perfectly acceptable candidate for becoming a Jedi. Add to all of this the fact that it is never mentioned that Anakin's children would be "chosen ones" in the style of their father (whatever that means), thus making their waiting pointless, as they should just finish the job themselves.
Anakin is described as the best starpilot in the galaxy in the OT, but no mention is ever made of this in the PT.
Technical Problems:
In the PT we see that a tracking device is a small, handheld device which can be hidden practically anywhere on a ship in order to track it, which makes Han's certainty that they are not being tracked in the OT rather perplexing. Before this I had assumed that a homing beacon would be a large device like the scanner suite that was hauled on-board the Falcon.
The chances of Chewie being familiar with Yoda from the PT and then meeting Obi-wan and company twenty years later strain credulity and shrink the universe immensely.
Anakin building C-3PO. In fact, having R2 and C-3PO in both trilogies is far too convenient.
Vader routinely says "The Force is with _", a practically meaningless statement considering that according to the PT, the Force is simply a certain concentration of organisms in a person's blood.
Force lightning can be blocked by a lightsaber in the PT, drawing undue attention to the fact that Luke threw it away in the OT, and thus exposing himself to this attack.
Thematic/Miscellaneous Problems:
The technology of the PT is much more advanced than in the OT.
Tatooine seems to be the center of the Star Wars universe, being in every Star Wars movie except one.
The Sith are focused on in the PT, whereas they are never mentioned in the OT. Conversely, destiny is focused on in the OT, where it is almost never mentioned in the PT.
When watched in numerical order, the three big reveals of the OT (Yoda, Anakin, and Leia) are ruined in the PT.
And perhaps the biggest problem of all: The central problem of the OT is that Luke cannot kill his father, whom he knows was a good man. From the PT, we know that Anakin murdered a tribe of Sandpeople and all the younglings in the Jedi temple, for no other reasons than 1: he was angry and 2: he was following orders in order to save his wife, whom he kills anyway. From this I can't call Anakin a good man. I would have no qualms with killing someone who did such evil acts, and thus if Luke knew the truth as the audience does, he probably wouldn't either. Therefore according to what the audience knows, the PT destroys the central conflict of the OT.