The main difference is in the role of the courts when it comes to applying the law; in common law, the judge is essentially the highest authority when it comes to enforcement. Unless another law specifically overrides a judge's interpretation of a law, they can make whatever decision they want and it will become "case law". Case law is a sort of pseudo-law that common law judges create when handling cases: it means that if a similar case is presented to the court, a judge can be required to interpret the law the exact same way. (This is called stare decisis.)

The advantage of this system is mostly that you very quickly end up with a giant body of legal text and any edgecases to your laws get hammered out very quickly and consistently. It allows for a more populist lawmaking system, since all the fine details are sorted out by the courts. It frees up politicians from considering how their laws will actually be enforced, since after a law is passed, the only effect it can have is on their political history. It also works well if you have a colonial empire where laws could take several months to spread across the empire, but you needed decisions to be taken in the here and now - common law moved some of those complexities to the local judges, where unless they'd be curtailed by "up high", they could be more practical enforcers of the law.

The disadvantage of this system is that you get a very large body of legal text fairly quickly. Actually interpreting common law tends to net you a quagmire of legal precedent that can often be more complex than the law itself and because there's a lot of it, you end up with the situation where a case can have multiple kinds of contradicting case law applied to it (in which the most recent version would "win", or the version decided on by the supreme court). It can turn legal arguments less into "interpret the law" and more into "throw enough citations at your opponent to force the judge's hand". A related problem is the risk of setting bad precedent: it's possible to craft "the perfect case" for certain legislation, leading to undesirable precedent being created, even if the law is correctly interpreted for that situation. This happens very often in the US, and is why Americans in my experience have no problem defending actively horrible cases for their one-issue cause, because they're more terrified of the precedent it'd set for the future. (The Internet Archive lawsuit is one such situation, where most Americans were more terrified about the precedent it set for copyright rather than a fair judgement on what IA actually did.)

Civil law on the other hand has no innate concept of stare decisis. Laws are as a rule only interpreted as they are passed. This doesn't mean case law doesn't exist, but it's not capable of forcing a judge's hand if the judge can find an argument to not follow it and the decision of one judge can't have ramifications down the road - you need multiple judges across multiple cases to find similar conclusions before it becomes case law. (This is called jurisprudence constante.)

The main advantage is that the total amount of legal text you need to understand the law is much smaller: if you get a law book, it'll just outright contain the majority of things you need to know about the law. There's still the usual "legal definitions don't always follow the common definition" going on, but that's not nearly to the same degree as happens with common law. It also forces the hands of politicians to consider much more about how the law is going to be applied in the future; a badly written law can have a lot of unwanted side effects, making it an easier target to strike from the books (which also tends to be more important in civil law; common law tends to leave bad laws on the books if case law happens to overturn it, which can lead to problems if the case law overturning the law is overturned). It gives "administrator" style politicians more leeway, since they need to actually interact with the bureaucracies that are going to enforce the law to make it work. There's also no "perfect case" situation of creating undesirable jurisprudence on accident. Judges don't have to consider the broader ramifications their decisions can have in those cases either, since they aren't writing new laws by deciding on those cases.

The disadvantage is that what you gain in comprehensibility, you lose in consistency. A judge can just outright decide to interpret the law in a different way from what's written and the recourse you have is pretty much to appeal it, demanding a retrial. This gives greater flexibility on "edge cases" where the law isn't super clear (without a worry for the greater ramifications that deciding on an edge case can cause), but can also lead to what should be an open and shut case getting a weird judgement (although the impact is fairly small ultimately - this is for example why the Hamburg Regional Court can keep farting out the most braindead copyright decisions that go directly against other laws and legal interpretations, but there's basically no downstream effect). Lawmakers try to prevent this by including "recitals", which are basically written addenda that try to clarify what each section of legal text is trying to address. This leads to the weird situation where the effective total body of law is smaller, the legal texts themselves tend to be much larger once you take recitals into account.

Personally, I prefer civil law over common law because of the lessened impact a bad decision can have.