For instance, sending information back (and possibly forth) in time might all of a sudden become a practical problem rather than a theoretical one.
Sorry to take the magic out of it, but that's not the case.
Neutrinos do not travel back in time, no matter how fast it turns out they were moving.
The principles of time-travel in relativity are due to a lack of a single fixed reference frame- which are the same principles that set c as the universal speed limit. It is in the nature of space-time as a single substance.
IF this turns out to be true, it means one (or more) of several things:
Something is effectively reducing the speed at which photons propagate (and perceived passage of time, since the electro-weak force is involved in material processes) that isn't affecting the neutrinos, or is not affecting them as strongly.
Cosmically, neutrinos do travel slower than light (such as from distant super novae); this may be a matter of context.
The neutrinos are not traveling faster than light if you don't factor in the space-time dilation caused by Earth's local gravity field.
That is to say, gravity may be slowing down time for light, but not for neutrinos, thus giving them an apparent relative boost.
The important consequences of that would seem to be that, yes, Einstein would have to be
partially wrong.
Special relativity would have to be thrown out, along with the inherent coupling of space and time, because in this case Neutrinos would provide evidence of a special independent frame of reference.
We would then have to offer alternative explanations for gravity, the perceived relativity of light speed, and time dilation.
It's relatively easy to compose working theories that fit the observations based on quantum mechanics, and I could go on to do that, but I'm already a little off topic.
The point is, that with a special frame of reference, space and time are independent of each other, special relativity is wrong, and velocities faster than c do not result in time travel.
Likely all this would do in the near future is give us a more accurate picture of gravity, solve UFT, explain dark matter and other Astronomical quirks, give us an irrefutable model of the start of the universe... which is nothing to sneeze at, but pretty much just academic. I wish I could say it would result in world peace by unifying the belief systems of all people in science... but that's not going to happen any time soon.
I would say, relevant to everyday life, it would probably give us a small leg up in quantum and optical computation resulting in profound annoyance to cryptographers. I'm not sure if this is a good thing or not, yet. Are we ready for post-quantum cryptography?
What I have to wonder, though, is what does quantum computation offer gaming? Maybe better sorting methods? I'm not much of a programmer, so I'm not sure.