What is the shared word origin and relationship between the tarantula spider and the tarantella dance?

There is an Italian connection.

In the Italian province of Taranto, Apulia, the bite of a locally common type of wolf spider, named "tarantula" after the region, was popularly believed to be highly venomous and to lead to a hysterical condition known as tarantism. This (type of dance) became known as the "tarantella".

https://en.wikipedia.org/wiki/Tarantella

Hilaire Belloc wrote a poem which used to be thought of as in the rhythm of the dance. However, I could find no reference to this, so it is obviously no longer believed. However, it does have a fascinating rhythm.
Tarantella
(1929)
Do you remember an Inn,
Miranda?
Do you remember an Inn?
And the tedding and the spreading
Of the straw for a bedding,
And the fleas that tease in the High Pyrenees,
And the wine that tasted of tar?
And the cheers and the jeers of the young muleteers
(Under the vine of the dark verandah)?
Do you remember an Inn, Miranda,
Do you remember an Inn?
And the cheers and the jeers of the young muleteers
Who hadn't got a penny,
And who weren't paying any,
And the hammer at the doors and the Din?
And the Hip! Hop! Hap!
Of the clap
Of the hands to the twirl and the swirl
Of the girl gone chancing,
Glancing,
Dancing,
Backing and advancing,
Snapping of a clapper to the spin
Out and in --
And the Ting, Tong, Tang, of the Guitar.
Do you remember an Inn,
Miranda?
Do you remember an Inn?
Never more;
Miranda,
Never more.
Only the high peaks hoar:
And Aragon a torrent at the door.
No sound
In the walls of the Halls where falls
The tread
Of the feet of the dead to the ground
No sound:
But the boom
Of the far Waterfall like Doom.