The advantages of using Julian dates (and what to do with the "- 2450000" part) have been described well in previous answers but if you're curious about the "barycentric" part here's an example.
Let's say I'm observing an eclipsing binary to measure the period which might be changing. Six months later I observe the system again. The measurements could be off by up to ~15 mins (the diameter of the Earth's orbit in light minutes) because I'm on the other side of the Earth's orbit now. Astronomers started using "heliocentric" Julian dates (HJD) to take care of the problem - as if the observatory was located in the center of the Sun - so the location of the Earth in its orbit was no longer a source of error.
HJDs weren't a perfect solution; the Sun doesn't sit in the middle of the solar system without moving. The Sun, planets, and all of the objects in our solar system are orbiting the center of mass of the solar system which is called the barycenter. Now Julian dates are corrected as if the observatories are located at the solar system barycenter instead. The difference between BJD and HJD is much smaller, around ± 4 secs.
Whether the differences is important or not depends on what is being measured. In the paper you referenced, data were being combined from multiple sources measuring very small timing variations and using BJDs made the timing errors as small as possible.