Given the huge difference between the books, I'll break it down into categories; Tone, Length, Audience, Structure and Canon.
Tone
The original Alan Dean Foster novelisation was intended for a young adult audience (14+) whereas the new Alexandra Bracken book is a junior novelisation targeted at 9-12 year olds, with far simpler language, lots more 'inner voice' (to explain what's going on) and substantially less graphic content. Let's compare the older with the newer;
Within the armor his head turned slightly, directing his voice to the tiny condenser microphone. “Here she is,” he called to those behind him. “Set for stun fore—”
He never finished the sentence, just as he would never receive the hoped-for commendation. Once his attention turned from the girl to his communicator her shivering vanished with startling speed. The energy pistol she had held out of sight behind her came up and around as she burst from her hiding place.
The trooper who had been unlucky enough to find her fell first, his head a mass of melted bone and metal. The same fate met the second armored form coming up fast behind him. Then a bright green energy pole touched the woman’s side and she slumped instantly to the deck, the pistol still locked in her small palm.
Star Wars: A New Hope - Official Novelisation
vs.
“There she is! Set your weapons to stun!”
Leia wasn’t about to set her blaster to stun. She fired, hitting the stormtrooper out in front. He let out a sharp cry as he crashed to the ground.
“Watch it! She’s armed! Fire!”
The firing would draw even more attention. She’d have to run, find better cover to hold them off just a little longer.
But the moment Leia turned her back, it felt like she’d been tackled from behind by a Star Destroyer. The hit from the stun bolt took all the feeling from her legs, sending her slamming forward against the rough grating under her feet.
A New Hope: The Princess, the Scoundrel, and the Farm Boy
Far more explication of what she's feeling, far less blood 'n' guts.
Length
The new junior novelisation is dramatically shorter.
Old novel - 183 pages
New Novel - 123 pages (including approx. 10 pages worth of illustrations and concept art).
Audience
It's notable that the newer book is written with a far greater emphasis on Leia's part in the story. Frankly, she's more spunky, as evidenced by her numerous escape attempts. Leia fans (and younger girls) will like that she comes across as a much fuller character with extensive insight into her thought process and that she gets her own entire third of the book rather than appearing as a supporting character in a novel largely focused on Luke.
Structure
The old novel follows the film pretty closely, starting out with the Tantive IV, moving to Tatooine to follow Luke, then following Han, then switching back to the whole gang, etc etc. The book also includes some scenes that were ultimately deleted but is otherwise quite faithful.
The new novel, by comparison, follows the characters in order that they're mentioned in the title, starting with Leia (the Princess) and following her until the scene where Luke/Ben enter the cantina, then following Han (The Scoundrel) until just after Ben Kenobi's death, moving to Luke's perspective (The Farm Boy). This switch in third-person focus means that there are necessarily some entirely new scenes such Leia's near-escape from Vader's Star Destroyer or Han's perspective of Luke/Ben's training session.
Canon Elements
The original novelisation was largely based on Lucas' first draft screenplay. The film went through several more revisions (and edits) before it hit theatres which means that there are a number of elements in the book that conflict with the accepted (film) canon. For example, Uncle Owen and Ben Kenobi are brothers and Han shoots first.
By comparison, the new novelisation sticks very closely to the films but because of its unique structure (see above) includes some scenes that were never filmed but are now considered to be part of the canon. This includes, for example, that Leia and Vader had met before.
Which should I read?
Although it's a personal perspective, for my money the junior novels are far better suited to casual fans of Star Wars, younger audiences, female readers and those with shorter attention spans and/or those who like books with short, discreet chapters.
The official novelisation is a more sophisticated read but it can seem a bit too Luke-heavy, the other characters can seem a touch two-dimensional and for hardcore fans, the canon discrepancies can be glaring.
