LLC[1] mostly does what it says on the tin: it limits the liability to the company, not yourself or other owners. Without this any debt or judgement against your company can consume potentially most of your net worth (subject to bankrupcy, etc.). In practice for a freelancer this means you can limit your liability pretty strongly, if you are passing everything through as payroll the asset value of your company can be quite small, and it's value without you nil.
On the other hand various types of corporate insurance cover you for particular risks. Depending where and how you operate you may have to have them by statute or by practicality. So you may have to carry liability insurance by law if you have an office where people visit, or a contract may require that you hold E&O insurance up to a certain amount. In a way the latter is actually your customer protecting themselves from your use of a LLC. Without it, in the case of a settlement against you, you could easily just turn around and say "fine, the corporate account has $5 in it, here you go" and then fold up the company leaving them with no recourse. With E&O coverage for certain types of errors, they know they can get covered in a settlement up to a certain amount.
E&O covers you for particular failures in providing the service you are contract for. Say you were an electrician and did some work on a new building. They sue you claiming your work wasn't to code and caused them $100k in trouble with the city - you disagree. E&O insurance covers your court costs and potentially your settlement if it goes that way. It isn't going to cover you if you get sued for libel because of things you said about them, etc.
This is also why, for example, it may be hard/impossible to get underwriters for some software consulting. Because there are not professional standards groups that are well recognized and because potential damages from software can be difficult to asses (your script change cost us $10mm in AWS fees) insurance companies may not want anything to do with it.
So that leads to a third prong of protections which you didn't mention, which is you need to pay attention to your contract terms (and set them as much as you can). Depending on your situation this can range from easy to impossible, but can have a huge impact. For example, I've successfully added clauses to limit all liability to actual spend on previous 6 mo.
[1] this is hugely dependent on jurisdiction, particularly with single member variants.