Suppose humanity finds a way to travel near the speed of light without the limitation of mass dilation, but still lacking the ability to avoid space and time dilation. Using this technology, the exploration of distant planets could be achieved with a reasonable amount of energy, however there would be an interesting bifurcation of human evolution into various timelines based on how far and how fast they were traveling.
For example, a colony of 500 humans is sent from earth in the year 2100 to explore a planet that is 5000 light years away (the galaxy is about 100,000 light years across). With their newly developed light speed technology, they they reach 99.999% of c with an acceleration of somewhere between 1-2g. It would take approximately 50 years as viewed from the ship to reach their destination, whereas 5000 years will have passed on earth. Subjectively, the 500 colonists would exist in the year 2150, whereas earth humans would exist in the year 7000. By this time, earth humans would be vastly different, culturally, technologically, and likely biologically, whereas the original colonists would appear like a relic from the distant past. Whatever communication might occur between the two populations would likely be confusing and archaic.
An even more interesting possibility is that earth humans discover a way to bypass space and time dilation while the original colonists are still traveling to their destination. Lets suppose this occurs in the year 4000, and rather than waiting another several thousand earth years for the colonists to reach their destination, they decide to directly send a group of humans to the same planet. When the original 500 space colonists arrived, there would be a civilization waiting for them that had existed for thousands of years, likely with their own culture, technology, and biologic adaptations. They would receive the colonists as a time capsule of human evolution from a remote and forgotten past...
If there were multiple such colonization envoys scattered throughout the millenia with varying levels of technologic advancement and distances to travel, there would be a strange collection of human populations spread through the galaxy each representing a different time point in human evolution. Each would continue to march forward, existing within their own frame of reference and level of development, isolated from whatever changes were occurring in their parrallel human realities, creating a diverse adaptive radiation across the cosmos, isolated not only in space, but also in time.
Of course, there would likely reach a point when the most advanced (likely earth humans, who would continue to experience no time dilation), would play "catch up", sending delegates of vastly evolved and technologically superior humanoids to spread their way of life, and knowledge. How this would be received is an interesting thought...Would we diverge to the point of conflict? Would we be so foreign to each other that our fellow man would appear as aliens? Maybe war will be a meaningless concept by then...Or perhaps the more advanced societies would simply cut off communication from certain 'primitive' societies, letting them believe they were alone in the universe, allowing a window into the past that could be studied from a distance, like living archaeology. Little pockets of isolated, primitive humans on random planets all evolving separately from one another, awaiting a time when themselves can reach out to the stars.