First of all, I think that this civilization is not a lot different from ours: we just lack the ability of stopping the ageing process. If we could, I think the most probable thing is that birth would become very rare and we would somehow all become immortals. It would change a lot our very notions of death and life because it would suddenly become a matter of choice.
In this context I think the ability of deep-space travel is negligible from respect to the impact of conscious immortal life.
In the other hand, trying to deep-space travel knowing that neither us nor our close siblings would be able to reach the destination is a real challenge. Giving the fact that the closest star system (except from ours) with habitable planets is several centuries away at light speed, people who decide to board the vessel have to expect spending their entire life and those of their siblings within this vessel before landing anywhere. This is very tough and for a very random result. What if the planet is covered in water, what if we were wrong? Is the ship self-sustainable so it can go from star system to star system until finding something interesting?
Plus you have the risk of loosing the point. I think the people inside the vessel -- which is actually becoming a community -- will start to forget about their goal. 790
They will eventually wander in space and start to wonder the same way we do. You have no way to strongly connect the two communities because nothing can travel faster than light and you cannot transmit anything from one human expansion to the other or to the Earth.
So maybe everyone is leaving the Earth. In that case I expect that your question turns into: what would we become if we were living in outer space? Well, I think that we have a good idea of how not good it would be given our current knowledge and technology. We have to experience gravity so we have to generate it. We know that maintaining an angular speed is extremely difficult and except if we have a very very large vessel we would still feel the difference of gravity between our feet and our head. We have to find an energy source, either we embark a fusion reactor with a lot of fuel (which I think is very unlikely) or we program our travel to go from sun to sun so we can fill up our batteries.
Unless you accept to have something like cryogenics abilities. I think that deep-space travel is very unlikely to happen and people would be very reluctant to embark in such a unpredictable and incredibly long mission. 790