They're both residences for the blessed dead: you know that much already. They appear to be two separate visions of the afterlife for the great and the good, that became blended at some point, but were later separated out again. In the Homeric Odyssey we get the Elysian Fields, located in the far west near the Ocean, and ruled over by Rhadamanthys; in the Hesiodic Theogony we get the Island of the Blessed, ruled over by Kronos (the Titan). Another variant seems to be traceable to the lost 7th-century-BCE epic the Aithiopis, where Achilles goes to "the White Island" after his death, a real location located near the mouth of the Danube on the Black Sea. But that variant never gained much currency.

Already in Pindar (early-mid 5th century BCE) we get the first two versions being blended: he talks about the Island of the Blessed being ruled by both Rhadamanthys and Kronos. Later on, we get Vergil (probably based on a lost Greek model) moving Elysium to the Underworld, making it a particular part of the Underworld that happens to be reserved for nice people. Conversely, Lucian's satirical True History makes the Elysian Field a geographic feature on the Island of the Blessed.

The blending of these variants could well be earlier than Pindar. At any rate it goes back a long way. If it's hard to keep them straight, that's because the ancients weren't particularly interested in keeping them straight either.