Deep Impact and Armageddon
Munsters and Adams Family
Phantom and Black Panther 
Both launched the same year and coincidentally (?) both got cancelled after 2 seasons.
Dead Man’s Curve and Dead Man On Campus
Two films with the same premise. Urban legend - If your college roomie commits suicide, you get automatic A’s for the semester. Failing student attempts to murder Roomie, and make it look like suicide.
The Matrix (1999) and Dark City (1998). Maybe The Thirteenth Floor (1999) as well. At least as far as “guy discovers reality is being scripted”
In 1984, we had Breakin, Beat Street and Body Rockin. One year later there was Krush Groove and Rappin.
1492: Conquest of Paradise and Christopher Columbus: The Discovery
The Illusionist and The Prestige.
The Abyss, Leviathan and Deep Star Six all three underwater sci-fi (for various values of sci-fi) movies, all three released in 1989.
True, in that they’re both turn-of-the-20th-century stories about magicians. But The Prestige does a twist in the second half and turns into a science fiction story. I admit I haven’t seen The Illusionist, but from what I’ve heard I think it stayed more grounded… though I’m sure someone will correct me if that’s not so.
Two films that have a different tone but very similar plot and both came out in 1987.
Attractive teenage boy is lured by mysterious female who turns out to be part of a vampire gang. Boy joins them; initially this seems very cool and a lot of fun. Inevitable conflict ensues, mostly due to boy’s younger sibling back in the “normal” world. Final battle sees off the vampires and rescues the girl.
Near Dark and The Lost Boys.
The Lost Boys was way more successful and popular, but Near Dark was the superior film, I reckon.
One year in the mid-late 60’s two broadcast networks (I think they were CBS and NBC) each had a short lived sitcom about bumbling superheroes.
Once was called Captain Nice and the other was Mr. Terrific.
Both were in response to the popularity of ABC’s campy Batman. Buck Henry created Captain Nice, which starred a young William Daniels, and it was definitely the funnier of the two. My seven-year-old self, however, preferred the slightly more earnest Mr. Terrific.
We watched part one of The Paradise on PBS. I was certain that I was watching a recasting of Selfridge. Identical plot, location, big department store that the banks won’t fund, womanizing lead with a cheesy beard. Tedious, to say the least.
Braveheart and Rob Roy came out fairly close together. When a reporter asked Mel Gibson about it, he replied, “It’s either the Collective Unconscious, or corporate espionage.”
I presume we do not include the “blatant rip-off” TV series, such as the “wild and crazy frathouse boys” sitcoms launched by each network the September after **Animal House **was a huge hit.
Big, starring Tom Hanks, and a remake of Vice-Versa (based on a Victorian era novel) both came out in 1988. They both centrally involve involve a boy whose mind is magically transferred into a grown man’s body.
[Also, according to the Wikipedia entry on Big, three other “age-changing comedies”, all also came out at around the same time: Like Father Like Son (1987), 18 Again! (1988), and the Italian film Da grande (1987). I had never actually heard of any of those before, but they all appear to have involved a boy’s mind magically transferred into a man’s body. That is five movies on this subject within two years!]
A remake of Freaky Friday (based on a 1972 book) starring Lindsay Lohan, came out in 2003. In 2004 the film 13 Going on 30, starring Jennifer Garner appeared. Both involve a girl whose mind is magically transferred into an adult woman’s body.
In both these pairs of movies, one of them is a remake, based on a book, and involves a child and an adult swapping minds (or bodies, depending on how you look at it): Vice Versa and Freaky Friday. The other of each pair is (AFAIK) from an original screenplay, and each involves a child’s body magically transforming into adult form, with no adult mind being put into a child’s body: Big and 13 Going On 30.
Freaky is the word!
One of the most famous examples is unmentioned here and in the links:
Jezebel (1938) starring Bette Davis, and
Gone With the Wind (1939) starring Vivien Leigh.
The settings and stories are very similar; both won Best Actress. The release dates may be misleading, the short b/w film having a much faster production tempo than the color epic.
I’d call it a draw:
Antz (Dreamworks) vs Bug’s Life (Pixar) - I think Antz was the better movie.
Shrek (Dreamworks) vs Monsters, Inc. (Pixar) - I think Monsters was better but a lot of people liked Shrek. Call it a tie.
Shark Tale (Dreamworks) vs Finding Nemo (Pixar) - A clear win for Pixar.
Flushed Away (Dreamworks) vs Ratatouille (Pixar) - Two weaker entries in my opinion. Call it another tie.
Aside from the pair of Capote movies (Capote and Infamous) and the two Dangerous Liasions movies (Dangerous Liaisons and Valmont), none of those subjects have been the subject of dueling movies.
The Girl and Hitchcock both came in 2012, both about Hitchcock’s obession with his films and female leads.