The difference between civil and common law is an elusive thing. On the surface it is enormous, but the deeper you dig, the less you find.
Sometimes it's claimed it's about precedent, and indeed some civil law jurisdictions claim that they do not believe in binding precedent. But of course for a legal system to be at all useful, decisions need to be consistent, and the idea that you can achieve consistency by writing every detail in a code so that every decision logically follows from the code is bullshit; if that were the case, all civil lawsuits would be 100% predictable and therefore rational actors would settle them and the judges could all go on vacation.
The reasoning I once read in some Dutch first-year law course notes was along the lines of "we don't do _stare decisis_, but we support equal treatment in equal cases, and it would be unequal treatment to treat you in way X when we treated the other guy in way Y, so we're going to follow precedent, but not because we must follow precedent, but only to avoid unequal treatment." I suppose that it is true that digging up absolutely ancient judgments is a little bit less convincing in a civil law setting, especially if there are periodic recodifications so that you can simply toss away a 17th-century precedent by saying it was an interpretation of the old code, not the new one.
Some say the difference is codification, but as you point out, not all status in common law jurisdictions are just piles of unrelated acts: a lot of the time, they are organized as systematic codes that are amended just like civil law codes. And besides, civil law countries have uncodified case law, too. The section on torts in the French civil code, for instance, is incredibly terse, saying basically that if you unfairly harm somebody you must compensate them. But of course France has tort law just like England does. Interestingly, since French court decisions do not normally provide much reasoning aside from quoting sections of the codes, the details of that tort law get elaborated mostly by law professors in books and articles; but in other civil law jurisdictions, like Germany, judges write long, reasoned decisions just like in the US (except more stilted and formulaic in style). And Scotland, considered traditionally a civil law country, has lots of English-style uncodified legislation.
So maybe then it's the Roman basis? Nah. English law had lots of Roman influence, too, and continental law had lots of influence from local customary law, canon law, and the law merchant. (Just read Berman to find out the details.) Maybe the continentals were bigger on pretending that it was all Codex Iuris Civilis all the time, but nobody ever really believed that.
So then what? Sometimes you hear particular doctrines called out as being significantly different, like consideration in in common contract law as opposed to the intention to be bound in the civil law of obligations. But the consideration rule has so many exceptions that if you can reasonably be thought to have intended to be bound, you'd better know the law very precisely if you still want to get out from under things based on lack of consideration. Besides, consideration may not be required in the civil law, but a payment can serve as evidence of a nonwritten contract.
That's not to say that there are no differences, but it's hard to pin down anything that really applies in all civil law jurisdictions and no common law jurisdictions or vice versa. Notaries, I think, are a pretty consistent difference, although they don't exist in some Asian civil-law jurisdictions.