In The Tempest, Shakespeare makes two references to Neptune, the Roman god of the sea.

In act 1, scene 2, Ariel describes to Prospero how he followed Prospero's orders to the letter, raised a tempest, and caused havoc aboard the sailing ship carrying Prospero's brother, Antonio, the King of Naples, and others who Prospero intended to shipwreck on his island.

ARIEL: To every article.
I boarded the King's ship; now on the beak,
Now in the waist the deck, in every cabin,
I flamed amazement. Sometime I'd divide,
And burn in many places; on the topmast,
The yards and bowsprit, would I flame distinctly,
Then meet and join. Jove's lightning, the precursors
O'th’ dreadful thunder-claps, more momentary
And sight-outrunning were not; the fire and cracks
Of sulphurous roaring the most mighty Neptune
Seem to besiege and make his bold waves tremble,
Yea, his dread trident shake. (1.2.229-240)

Ariel says that he appeared as a burning fire in several places on the ship, then as one big fire. He seems to describe St. Elmo's fire, an awe-inspiring electrical phenomenon that resembles a glowing ball of light that appears during thunderstorms on pointed structures, such as the prow ("beak") of the ship, and "the topmast / The yards and bowspit." (There is a remarkable scene of Captain Ahab "taming" St. Elmo's fire in the 1956 movie Moby Dick.)

Ariel says that his fire shined brighter than Jove's lightning before dreadful, deafening thunderclaps. He boasts further that he raised such a tempest in the sea that even the mighty Neptune was so frightened that his trident shook in his hand.

Prospero makes a another reference to Neptune in act 5, scene 1, near the end of the play, when Prospero vows to break his staff and throw his book of magic into the depths of the sea. Prospero inscribes a circle around himself and calls to the spirits who been loyal and steadfast to him and who have helped him with his magic.

PROSPERO: Ye elves of hills, brooks, standing lakes and groves,
And ye that on the sands with printless foot
Do chase the ebbing Neptune and do fly him
When he comes back... (5.1.38-41)

Prospero says that he's done all manner of magical things with the help of the elves who live in the hills and forests and the spirits who run on the beach without leaving footprints and run away from the waves ("Neptune") when they come back to shore.

All in all, Neptune really has nothing to do with raising the storm, except to serve as a frame of reference for Ariel's boasting of his prowess in raising a tempest. Neptune also serves as a metaphor for the sea when Prospero is praising and thanking the spirits who have helped him with his magic.