They can hear more, but not better.
Because of the density of water, vibrations are changed in pitch, and may be muffled and garbled. Since clarity is the most important factor in hearing, underwater communication is mostly done by hand signals instead.
On the other hand, sound moves more quickly underwater, and as a result you can hear from a greater distance. Things can be heard from a lot further away and/or are louder than in air. Unfortunately, because of the higher speed, and reflections from the bottom, sound direction is almost impossible to determine by ear.
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Sound moves faster through water than the it does in air. However, divers cannot hear better underwater because the human ear is not designed to hear clearly underwater.
In an air medium, a person can hear more clearly and from which direction sound comes from. A person can hear "direction" because the sound hits the nearest eardrum first, and there is a barely perceptible time lag until it hits the other ear drum. Thus the human brain is able to determine which direction the sound is coming from.
However in water, the denser medium allows sound to travel five times faster than it does in air. The water itself muddles the clarity of the sound diffusing the sound vibrations over a wider area. That's why you cannot hear clearly underwater. In addition because sound is faster underwater it hits the ear drums almost simultaneously, makes it very difficult to determine direction underwater.