This is really a matter of personal preference/linguistic "etiquette". Using the infinitive (which is commonly-- though possibly erroneously-- construed as a "subjunctive" in popular grammar teaching) tends to belong to more formal/learned usage. Of the two, I suspect it's the one that occurs less naturally in everyday usage, and for many speakers may well be used only once learnt artificially rather than being acquired naturally.
Some languages have special verb forms, consistently used and acquired by native speakers, that are used to mark a 'non-assertion' (essentially, something that an assertion that can be agreed/disagreed with). Such forms are usually termed "subjunctive". English actually used to have such forms, but no longer does. Speakers who use the infinitive in forms such as "It is important that John bring..." may be attempting to mimic the presence of a subjunctive form. A potential difference, then, is that in the first example, there are two assertions you can agree with:
"It is important that John brings his lunch to school."
Response A: "Yes, I know it is."
Response B: "Yes, I know he does."
whereas in the second case, only response (A) makes sense:
"It is important that John bring his lunch to school."
Response A: "Yes, I know it is."
Response B: "*Yes, I know he does."
In other languages with subjunctive forms, and in English in the past, speakers make this distinction fairly naturally. For example, in French, any native speaker will generally make judgements similar to the above on sentences such as:
Jean a dit que Marie est partie immédiatement.
(Indicative: "Jean said Marie left immediately.")
vs:
Jean a dit que Marie parte immédiatement.
(Subjunctive: "Jean said that Marie leave immediately", i.e. "Jean told Marie to leave", "Jean ordered for Marie to leave")
However, in contemporary English, it's not clear that this distinction isn't just an artificial invention, and one not intuitively made by all native speakers.