To be very specific to the question, no, they didn't "get Relatively wrong" per se, they intentionally thwarted our expectations based upon our understanding of Relativity as a plot device.
The conversation between Palmer and Ellie is used to set the expectations of the audience, that if Ellie successfully travels anywhere outside of our Solar System in a timespan we can fit inside a movie, she'll be leaving the Earth she knows and the people on it behind, and many decades (at least) will have passed by the time she gets back. When the opposite occurs at the end of the movie, when she experiences time dilation "in reverse" as it were, it's used as a reason to completely discredit her story and leave both the in-world population and the audience hanging on what really happened - was her experience genuine or was it some sort of stress hallucination brought on by the intense forces at work on her body? Because all the science we know indicates that if she had gone anywhere, we'd be the ones waiting.
They are very intentional about understanding what is suggested by Relativity and circumventing it to accomplish two main goals:
- Double down on the enmity between Bureaucracy (personified by James Woods) and Knowledge - in this case, showing the government going so far as to use scientific principles (Relativity) to hide the in-movie truth (that we can circumvent the limits we believe to be imposed by Relativity, and this alien tech is the evidence)
- Enliven a sense of wonder and imagination at what could be that we don't know yet
The rational explanations for how she would have accomplished her trip, wormholes, etc, are less important to the movie than why she did it. The other answers do a good job of explaining the how, I hope this helps to understand the why.