In Irish history Cromwell's reputation is due to his rather bloody Irish campaign that lasted from 1649-1650. Ireland was Catholic and held Royalist sympathizers, so Cromwell felt the need to invade and restore order their. Cromwell was no fan of the Irish, both because of their support of the Monarchy and their Catholicism. Add in that there were Irish attacks against Protestant settlers in Ulster. And finally just the general way warfare and sieges were conducted in the time period meant that the campaign was very bloody and featured a lot of very destructive actions. Most famously, during the siege of an Irish town, Cromwell set fire to a church that was being used a defensive point by some soldiers, which resulted in about 80 or so Royalists being killed. He then proceeded to sack the town, killing about 2000 people.

So there are still debates about how brutal he was, many argue that Cromwell's actions in Ireland were fairly standard for the time, and that he wanted to crush the Irish and the Royalists so that he wouldn't get bogged down in a guerrilla war. Still it was a very brutal conquest, and it naturally left quite a scar on the Irish conscious.

In English history Cromwell was actually not well liked for a large portion of English history. Mainly because he'd been a less than stellar ruler and his time in power saw economic issues (debated if they were truly his fault, but none the less) and a rather strict puritan rule cast over England.

His rehabilitation came in the Victorian era. When Cromwell was held up as a great protestant paragon fighting against a corrupt Catholic monarchy. He was turned into a figure that had fought in favour of parliamentary democracy, and the lower classes.

My view is that he lies somewhere in the middle. He wasn't a demon, the brutality displayed in his campaigns was strict, but fairly standard practice for the Early Modern Era. However, he dissolved the parliament of England forcefully and instituted a vastly curtailed version that was quickly dissolved and Cromwell became leader of England via his position as Lord Protector. So he can hardly be held up as a great friend of democracy.

As has been said already, he is certainly no more respected in England than any other Dictator. The Whig strand of History does tend to praise him as a progressive, and the Left similarly tend to view him in a good light as an anti-Royalist. But he is by no means an English hero. Irish nationalist mythology, obviously, portrays him as an evil destroyer, which is no more an objective truth than the Victorian Whig view of him as a hero.

If you look at him from the Irish side, it is not surprising. He was responsible for the most complete conquest of Ireland yet, destroyed the ruling elite and replaced it through a massive programme of land confiscation. He laid the foundation for the Protestant ascendancy which governed after the Restoration. He also was fervently anti Catholic. For Irish nationalists he was one of the key figures in the subjugation of the nation so it was inevitable that they would demonise him. In doing so they blamed a vast range of problems on him with little regard for nuance.

Whereas in England his reputation is complicated by his military achievements and the fact that he held the state together during the Protectorate, in Ireland there was little to recommend him. Even though modern historians have questioned the traditional view they have not found anything particularly positive about Cromwellian Ireland.