If a man gives his neighbor money or vessels to keep... [Laws of "shomer hinam"] (22:6-8)
If a man gives his neighbor a donkey, ox, sheep, or any beast to keep... [Laws of "shomer sachar"] (22:9-12)
These passages are the source for the halachic concepts of "shomer hinam" and "shomer sachar" - the consequences for unpaid and paid guardians of property when the property is lost, stolen, or destroyed. The consequences are, of course, more serious when you are being paid to be the guardian and nevertheless something goes wrong.
It's interesting that the verses themselves never mention whether or not the guardian is getting paid. In fact, there seems to be a clear, different distinction between the cases: the second case deals with live animals, the first with all other property. What then is the basis for deriving the concepts of "shomer hinam" and "shomer sachar" from these verses?
To answer this, we must first remember that parshat Mishpatim consists of "case law", in which specific applications of the law are used as examples from which the law in other cases can be derived by comparison. For example, if my animal kills your animal, I'm halachically required to pay you certain amount of damages. This is true no matter which species of animal does the killing. But parshat Mishpatim doesn't say "If an animal kills another animal", but rather "If an ox gores another ox". The reason is that oxen were the most dangerous domestic animals in the ancient Middle East. If one animal killed another, it was most likely a case of one ox goring another ox. Only this most common case is specified, and we are expected to determine the law for all other cases by analogy.
We must therefore ask about each law in parshat Mishpatim: is it intended to be self-contained, or is it just one example of a more universal principle? This determination should be made using common sense. For example, I see no justification for a special law for when my ox gores your ox, which doesn't apply when my sheep butts your sheep or my dog bites your ox. It's more reasonable that a single principle should apply to all cases of animal damage. Since that understanding makes the most sense, and since special laws for the sheep and dog cases are noticeably absent, I can tentatively conclude that the ox case was intended to cover all animal damage cases.
Let me try the same analysis with the two guardian cases. I, at least, can't think of any logical reason why a guardian should have to pay for an animal stolen from him, but not for other stolen property. One possible parallel is the convicted thief, who pays fourfold or fivefold if he sold or killed a stolen animal, but only pays double in all other cases of theft. But this seems intended to deter killing the animal (which would make the crime harder to reverse), and would be less relevant in the guardian case.
Perhaps a better distinction follows from the fact that guarding animals is unlike guarding other types of property. Non-living property can simply be placed in a vault and forgotten about until the owner wants it back. In contrast, keeping an animal requires constant effort. The animal must be fed, cleaned up after, taken for walks, watched for diseases, and so on. This is an extra commitment beyond a person's normal workload, and in the case of food it represents a significant expense.
Therefore, my guess is that while a person would willingly guard a nonliving object, it would take a substantial payment to get someone to guard an animal for any substantial length of time. Perhaps it can be presumed that animal guardianships were generally paid, while other guardianships generally were not.
With that assumption, the halachic difference between the cases is trivial to understand. Extra payment implies an extra level of responsibility. If the guarded object is stolen or destroyed, only the paid guardian is held responsible.
Now, this may not be the only reasonable understanding of the verses, but it's at least as defendable as any other I can think of. There is inherently a level of ambiguity in the "case law" method. When the principles are not stated, it's always possible that someone might disagree with me and assume the existence of a different principle than I do. This can be a disadvantage in that the laws are more ambiguous. But it can also be an advantage, in that there is more flexibility to sensibly solve complicated cases, without being anchored to an explicitly stated principle even when it leads to contradictions or unfairness.
Practically speaking, though, parshat Mishpatim is not detailed enough to give a conclusive ruling in many cases. That is why we had a single Sanhedrin to decide all cases for the Jewish people, and why we now have an "oral" tradition which records how these laws are in fact to be applied.
Whether the "correct" interpretation recorded in the oral tradition was specified at Sinai, or whether God intentionally left to us to interpret and apply parshat Mishpatim as we thought best, is a very interesting question to which I do not know the answer.