Something that I vaguely implied previously, but I don't think actually said, was that there is a difference between roles and cases (even worse, there are multiple things that "role" could refer to). Roles are, in theory, purely rational, language-independent categories which describe how nouns relate to their clause's verb. Cases, on the other hand, are language-dependent categories representing many things, and there is rarely (if ever) a 1:1 mapping of the two for a language.
The Grammer of Discourse hypothesizes at least ten universal roles, which I'll only briefly describe.
Experiencer: the person experiencing an emotion or sensation
Patient: the one an action acts on
Agent: the one willfully performing an action
Range: an extension of the verb, such as indicating how, e.g. "Your blood smells good"
Measure: an extension of the verb indicating how much, e.g. "I was only bitten a little bit" (these examples brought to you by Vampire Knight)
Instrument: something which is used to perform an action; this can also be used for animate entities who unintentionally perform an action
Locative: the location an action occurs at
Source: the starting point of some kind of movement or transfer
Goal: the ending point of movement or transfer
Path: the path taken during movement or transfer
If we were to compare this list of roles with typical use of the Latin cases, we would get the following. Note that this list is approximate, and some of the roles like measure and range I'm not even sure how to represent in Latin.
Nominative case: agent, patient, experiencer, instrument
Genitive: unrelated to role in the sentence (roles refer to relation with the verb, not with other nouns)
Dative: goal, patient
Accusative: patient, experiencer, goal, rarely source
Ablative: source, instrument, locative, goal, path, possibly range and measure (some of those requiring prepositions)
Locative (rare): locative
Vocative: not related to role
However, while case is language-specific, some themes (common cases) occur much more often than others. Of the Latin cases, the nominative, genitive, dative, and accusative occur very frequently in all languages; this is not surprising, as these seem the most essential to language in general (though note that they are not guaranteed to mean exactly the same thing in all languages).
The nominative case is roughly defined as the subject of the verb. For transitive verbs having a direct object, the subject is the one performing the action (e.g. "He poked her"); for intransitive verbs the subject is the single argument (e.g. "He was hit"). The accusative case is the object of transitive verbs. Any language having this structure is called a nominative-accusative (or sometimes just accusative) language (which we're going to call N/A in the rest of this post).
However, two others - the ergative and the absolutive - also occur very commonly in languages. The ergative case is defined as the subject of transitive verbs. The absolutive case, however, includes both the subject of intransitive verbs and the object of transitive verbs. Languages using this system are called ergative-absolutive (or sometimes just ergative; E/A, here).
At first this seems very strange and arbitrary - splitting the subject depending on whether the verb is transitive or intransitive. However, this is due to the fact that we don't speak an language. In fact, even the word 'subject' reflects this bias in thinking. The N/A split carries the paradigm that all actions are done by somebody/something, regardless of whether the action is intentional or unintentional, or even whether there's anyone performing the action at all (e.g. in "He fell"). This is called the subject, and for transitive verbs, the one acted on is the called the object; thus the N/A split actually corresponds to a subject/object division.
However, we get a different picture if we discard this assumption and look at things from the perspective of roles. In reality, with many intransitive verbs (such as the one shown above) the "subject" is not the one doing the action at all, but rather the one who is subjected to the action - the patient. Thus the E/A split is based on the paradigm that the ergative case is the doer (agent or instrument) of the action, while the absolutive case is the patient of the action - an agent/patient separation. Taking it one step further, some E/A languages even require that the ergative argument commit the action intentionally, and use a different sentence structure to indicate otherwise (e.g. split-intransitivity languages use either the ergative or absolutive case for the subject of intransitive verbs, depending on whether the action is intentional or not; others use the passive voice for unintentional actions; etc.).
Given this, both seem equally sensible, and the choice itself now seems arbitrary. It's worth noting, also, that most languages in the world are either N/A or E/A. Languages using other systems are rare, which might suggest that the N/A and E/A splits are more sensible and/or useful than other methods. But hold onto that thought.