Deep healing: Delta brainwaves are known to be very restorative. Their regenerative effects are very significant. Delta is the brainwave that helps restore your body back to its natural state.
Meditation & Yoga: In some forms of Yoga, meditation and Advaita Vedanta ("non duality"), being aware in the "dreamless sleep" state is considered the highest state of consciousness.
Headache & Pain relief: In a test done in 1985 by Solomon GD, it was shown that 14 out of 15 headaches went away with the use of Delta brainwave entrainment.
Improved sleep: In a 2002 study it was proven that 58% of the participants got improvements in sleep and reduction in fatigue using Delta brainwave entrainment.
Anti-aging: Delta waves release various hormones that promote anti-aging processes. These hormones include DHEA as well as melatonin.

The Delta brainwave frequency range (0.5 to 4 Hz) is so deep that they are considered deeper than dream state, is found as the dominating frequency range in deep, dreamless sleep and in extraordinarily deep meditation where awareness is fully detached. In meditation, Yoga etc, the Delta state is usually considered the "final stage" to master.

Some people who do Yoga Nidra (a type of deep sleep meditation) are able to remain conscious while in Delta-sleep. In for example Advaita Vedanta ("non duality"),  being aware in the "dreamless sleep" state is considered the highest state of consciousness.

Apart from the "spiritual" benefits of Delta brainwaves, the health benefits from Delta brainwaves are incredible. They are linked to Intuition, anti-aging, improved sleep, cell protection, Human Growth Hormone production (HGH), improved immune system, reduced anxiety and headache relief, just to name a few.