I've heard the explanation that the Enterprise from 2009's Star Trek looked so much more advanced was because data gathered from the Kelvin initially scanning Nero's ship when it first appeared enabled them to upgrade their tech. I'm wondering if they've ever detailed exactly what technologies were advanced or enhanced or discovered.
First of all, here's a quote from J.J. Abrams himself confirming the impact of the scans of the Narada:
The idea of the story is that at the beginning of the film something happens that changes the course of history. They cross paths with this futuristic ship, and it changes everything that would've been the classic series Trek fans are familiar with... One could argue that, based on the readings they got from the [Romulan] ship that showed up, it inspired ideas and technology that wouldn't have advanced otherwise... On the one hand, you could answer the question by saying that history got a boost, an adjustment, from this moment at the beginning of the film...
Now, as for your actual question, one thing that seems to have changed due to the Narada encounter is the size of Starfleet's technology:
The Narada was massive in comparison to the Kelvin, and the Abrams NCC-1701 is massive in comparison to the Prime Timeline's NCC-1701.
Now, there are two ways to interpret this:
- Starfleet engineers might have reckoned that the battle went badly between the Kelvin and the Narada because of the sheer size of the Narada, and this may have instigated an upward trend in the size of Starfleet starships that was more rapid than in the Prime Timeline (the Vengeance in Into Darkness is further evidence of this "bigger = better" approach to battle readiness); or
- the Abrams NCC-1701 is filled with larger tech that requires more space to house.
It could be a combination of both.
As to which exact technologies within Starfleet vessels were influenced by the Narada data, there are no canon (or even quasi-canon) statements regarding these.
It might be possible to extrapolate what they must have been from what we see in the Abrams NCC-1701. I am hesitant to say that improved display screens and console interfaces are some of these technologies, mainly because I would guess that such things are not high priority when scanning a hostile vessel. Also, these improvements could simply be a knock-on effect of having more time to pursue research into ergonomics because of the problems that may have been solved with regards to propulsion and weaponry due to the new data.
I would like to say that weapons systems on the Enterprise have been enhanced as a direct result of the encounter with the Narada. However, the lack of descriptive statements regarding the on-board weaponry limits our ability to make definitive statements of this kind.
Also, the transwarp beaming device that Khan uses in Into Darkness was the result of Spock's interference, not of the Narada's!
We can, nevertheless, take note of the following items from Into Darkness :
- The experimental torpedoes developed by Section 31 (in which Khan hid his followers) could have been influenced by the Narada's torpedoes — plenty were lobbed at the Kelvin.
- The Enterprise and the Vengeance made the trip from Qo'noS (in the Beta Quadrant) to Earth rather quickly, suggesting improvements to propulsion that may have been instigated by scans of the Narada.
We can also try to link the formidable phaser cannons on the Vengeance to the Narada scans — phaser cannons had previously not been seen on Starfleeet vessels in the 23rd Century — but they are much more likely to be the result of Khan's influence, as Khan's job was to make the Vengeance as fearsome as possible.
This is probably the best we can do without specific canon-based indications.