It's an old Q, and completely answered by @TomCotton from the point of teaching/learning the Lating language, but I think it deserves a look from another angle, a purely linguistic rather than a philological/didactic standpoint, because the most important syntactic feature has not been mentioned (common to all IE languages that have retained both the case and the preposition):
- It's always incorrect to analyze syntax of a nominal case (that of a noun, adjective or a noun phrase) as such when it forms a prepositional phrase, because its case is governed by that preposition.
mensa in your in mensa example is in the Abl. not because the case is syntactically meaningful, i.e. chosen among other cases to convey a specific meaning. It's in the Abl. because the preposition in requires it to be. The unit of syntactic analysis in this case is the entire prepositional phrase (you'd often see the acronym PP in parse trees). And you cannot meaningfully ascribe it a property of having a case at all. The preposition makes the case irrelevant (even if it's one of a very few prepositions, chiefly of location/direction that may govern two cases, such as in, supra, sub, etc.). The prepositional phrases with the Abl. nearly totally absorbed the Loc. and Ins. cases in Latin. It hardly useful to think of these as having anything to do with the Abl. proper, originally the case of separation, and by extension source, cause etc.
Cf. Romae, which is in the Loc. case (the Abl. would be Romā), retained in the Classical language only for place names in certain contexts, and a couple of frozen words, like domi and humi. This shows that the PP, e.g., in Hispania, has essentially a locative force, despite the noun taking the Abl. form, otherwise expressed by a prepositionless Romae.
As for the "ablative of attendant circumstances," the syntactic analysis gets mushy. First, this is originally a function of the Instrumental case, absorbed in Latin and other Italic languages by the Abl. Second, the AA, being an adverbial modifier, may and is often to express "attendant circumstances," just like a subordinate clause coordinated by cum, originally a conjunction (quum). Third, the AA itself has probably developed from this usage by dropping the conjunction turned a preposition.
Modern teaching of Latin (in the sense of learning the language) follows the extensive and inexact classification of the use of the Abl. and other cases, harks back to the ancient grammarians, and is at odd with the modern semiosyntactic analyses. Linguistically, when studying the language (as opposed to learning to read/write/speak it), it is more fruitful to look at its diachronic development w.r.t., speaking of the question, its case and preposition systems, which are most often in a conflict with each other. In modern Romance languages, with Romanian being probably a single exception, prepositions won over cases entirely.
The AA is a later development, and it can often be replaced with subordinate clauses with cum (originally Instr.) or dum (originally Loc.)