Having watched Episode VII, it can't escape anyone that it is very similar to Episode IV in terms of plot.
My question is, why is there such a similarity in the storylines?
In response to mounting criticism that The Force Awakens is a rip-off of A New Hope, director J.J. Abrams defended his creative choices by saying:
"I can understand that someone might say, 'Oh, it's a complete rip-off!'...What was important for me was introducing brand new characters using relationships that were embracing the history that we know to tell a story that is new — to go backwards to go forwards."
(Source — J.J. Abrams Responds to "Rip-off" Criticism)
Basically, he felt that the key goal for this film was to introduce new characters, and that the best way to do this was by "using relationships that were embracing the history that we know" — some kind of muddled public-relations-speak for
"We borrowed a lot of structure from A New Hope so that we could introduce new characters using a formula that works (Episode IV) rather than one that doesn't work (Episodes I—III)."
(Translation — Me)
I am so glad you asked. I have been debating this with a friend.
The two movies both have familiar elements, but the plots are not the same, and are hardly any more similar than any other two movies from the saga.
Plot refers to the sequence of events inside a story which affect other events through the principle of cause and effect. Source
Jumbled up details that are similar do not make the plots match. The introductions of both ANH and TFA are opaquely similar, as well as the destruction of an ultimate weapon near the end, but as far as the actual plot is concerned, that's the end of the similarities of the sequence of events of the narrative. The very beginnings and near ends are similar, but between those and the bitter ends are wildly different.
The similarities of many details are curious, but the differences in generic story plot are massive.
In ANH, key to the plot is a couple of droids paired together. Though the same droids appear in TFA, there is no companionship, no journey. BB-8 is sent off alone and takes no partner.
In TFA we see a stormtrooper battle first hand. In ANH all we see on the screen is evidence of a battle, and until Obi-Wan reveals the perpetrators, no one else knows what happened to the Jawas. There is no realization that our hero's parents have been murdered, certainly not in the first act of TFA. Later, Rey is guided to understand her family (we assume) is not ever returning to Jakku. She does not have the devastating experience of seeing their charred and smoking bodies.
The revelation that both Ren and Vader wear black doesn't squeeze the PLOTS any closer to each other.
No one steals any spaceships in ANH, yet it happens a couple times at least in TFA, and is essential to its plot.
The cantina scene in ANH is in the first act, yet a similar scene appears in the second act of TFA.
One of the important plot details in ANH is getting stuck the belly of the whale, the trash compactor on the prison level, and how the characters escaped being sandwiched, which is entirely missing from TFA, though we are reminded of it in passing. The similarities in plot, cause/effect in sequence, by the middle of the first act begin to diverge, and TFA sprinkles reminders of the other movies, but the plots are considerably different.
What character in ANH does Finn correspond to? Who is his counterpart and where are his two lightsaber duels (one with a storm trooper, one with Ren) in ANH? These events are important plot details to TFA and you will not find them in ANH.
In TFA, an important character, Poe Dameron, presumably the counterpart to Leia in ANH, is thought to have been killed in a crash, only to appear triumphantly alive later in the film leading a rescue assault. This does not occur in ANH.
There's a lot of Obi-Wan on screen apprenticing Luke in ANH, but this structure simply does not exist on screen in TFA, it is only inferred.
A plot device unseen in ANH is used extensively in TFA, the Vulcan Mind Meld, or force mind-reading. Though in Episode VI, Darth Vader gleams some details from Luke during their lightsaber duel concerning Leia's true identity (thanks, Wad Cheber!), why does Luke have to go through the extensive physically and mentally intensive training with Yoda on Dagobah when it would have been far simpler to pass that knowledge and skill telepathically? There is no such thing in ANH, or at least it is hardly, if ever, used in that film as a plot device to progress the story along.
The hero in ANH, Luke, has no lightsaber duel.
Yoda does not appear in ANH, yet in TFA we meet a Yoda counterpart in Maz Kanata, a force sensitive.
There is a tremendous and curious plot device used at climax of the film, the end of Rey's lightsaber duel with Ren, forcing them apart. No where in ANH can this or anything remotely similar be found.
A father-figure being killed, all while the main protagonist watches helplessly from afar... sounds familiar. But it doesn't hold up to scrutiny. In ANH Obi-Wan's death is true purposeful sacrifice to allow the others to escape. In TFA, Han is motivated by something else, something very different, as a parent and not as a mentor, and his action is not a meaningful sacrifice. Instead of the sacrifice of a Christ-figure, Han is a victim of deception and patricide. A Christ-like figure appears in TFA, but not until the final scene.
There is nothing like the conclusion in TFA in ANH... its so different that it can't even be intelligently compared, only contrasted. In ANH, our heros are publicly rewarded with medals. In TFA... well, you know... what a look! But if there is a similar sequence in TFA, it happens at the beginning of the 3rd act, "the end of the Republic," mean speech and applause.
What is clear is that much dramatic theater has identical framework. Most stories have an antagonist, a protagonist, and other characters, and are told in three acts, with an exposition, rising action, climax, falling action, and a conclusion. Creepy how all movies are nearly identical!
When you take similar characters, and similar settings, and similar actions, you end up with something that is similar, but unless the sequence of events, the motivations, the cause and effect is identical, the plots between two stories will not actually match.
Two plots cannot actually match unless the sequence of all events is also matching, point for point. And for TFA and ANH, this does not occur.
So since no one else mentioned this here's the likely answer: its the hero's journey, a mythological epic archetype noted by American mythologist Joseph Campbell. Lucas credits Campbell as an inspiration to the plot of Star Wars, and Campbell is both directly influential in pop culture story arcs (he's pretty well known) and indirectly (he used classic source material familiar to many like Gilgamesh, Homer, various Eastern classics, etc. meaning others arrived independently at the same archetypes).
This trace back to the work of Campbell also explains why movies/books/video games are constantly accused of 'ripping off' the plot of Star Wars1, as if Lucas (or Campbell really for that matter) came up with the idea. Some (again, notably Campbell) argue that there is something intrinsic to our humanity in the epic hero. So the plotlines of the two movies are similar (as well as pretty much every other epic narrative) because they draw from the same source material.
1Examples: The Sword of Shannara, Final Fantasy XII, pretty much any fantasy fiction rags-to-world-savior arc.