Basically anytime anyone does anything with Steem and ends up with a result they don't expect. When I was with Minnowbooster, this was "I sent a transfer and didn't get a vote" or "I got the wrong size vote." (Even today Steemworld is still the only place it's reasonable to see a transfer and a vote on the same page.) At SBI it's "I sent a transfer that was missed" or "The bot isn't voting my posts" or "Is the bot working right today?" Reading a vote pattern anywhere else to figure out which part of the bot might be broken is just lol.
In the Mesopotamians I used Steemworld as a list of posts the accounts were voting in order to generate statistics and do the membership curation posts. At that time no front end had the ability to create a feed of posts an account had voted on.
When I was writing test code for curation maximization, SteemWorld's curation return predictor was essential for being able to work on my methods in real time. Nobody else does this.
At Otterworks, without SteemWorld I might not be able to develop for Splinterlands at all. Not only is it there for me to verify transactions that go astray, and for the many, many times when Steem-Engine can't report its own data correctly, but the other explorers have all decided to obfuscate Splinterlands custom JSONs. The data structure in them is both extremely important and almost entirely undocumented, so pretty much the only way to learn to do anything is to make a transaction and peer at its contents. Even on Steemd they look like this:
That's not expandable, by the way. If something goes wrong with one of those it's colossally useless. On Steemworld they expand to their full data and I can see what's going on.
Lately at Herons Unlimited I've been having to create accounts with account creation tokens due to a technical issue with a new way Splinterlands is doing signups. Even as a dev, it is ludicrously difficult to do this in a practical fashion anywhere but SteemWorld.
Yes, it's a UI for the Steem API. That's something everybody doing anything here needs, and it's the only one that does it well on an immediate scale. Beem's probably more important to me overall, but Beem is clunky as hell to do anything you need to do one-off, like answer "what went wrong in that transaction?" The other web interfaces aren't worth talking about; all the ones that have decent UI, like Peak's history, selectively omit things. As I posted above even the canonically rawest block explorer made itself useless for a lot of the transactions I deal with. Steemblockexplorer is ok once you're used to its quirks, but it still takes fives times as long to do anything with.