Jack Sparrow has some unique content, as well as early/midgame difficulty spikes (see: missile strikes). Jack Sparrow is the almost-impossible meme difficulty, not the intended experience. It's aimed at experienced players using some powergamer stratz; for example if you don't get base defenses online within a few months the aforementioned missile strikes will quickly put your run into a deathspiral.
I usually play on Blackbeard, I find it's pretty fairly balanced. The easiest difficulty, Capn' Kidd, has the Wasteland Sorceress event disabled which locks you out of voodoo content. John Silver, between them, is good for noobs and I use it for messing around and casual runs. If you're looking for a challenge without the meme difficulty then Davy Jones is good.
The difficulty levels increase the number of enemies spawned in missions, as well as the percentage of enemies which start facing your landing zone. It also decreases the cash you get from selling items, so the profitability of loot is decreased and the viability of various manufacturing-for-profit changes.