The difference between formal and informal registers lies in the rules which are followed, not the medium of delivery. They are for all practical purposes different dialects of the language.
Formal utterances follow the rules which obtain in written academic works. They characteristically strive for careful construction with complex subordination, close attention to structural and rhetorical effect, and careful choice of words for unimpeachable precision. (This is not to say that these utterances always succeed in these laudable aims; but they try.)
Informal utterances follow the rules of ordinary discourse. They are characteristically marked by relatively simple syntax and mostly coordinate construction; they employ words more loosely, with greater concern for popular than academic use, and they tolerate a substantial amount of grammatical imprecision which is not permitted in formal utterances.
Informal does not mean 'non-standard', although it may embrace 'non-standard' uses for rhetorical effect. When we speak of Standard English, we mean utterances appropriate to their context and audience, not utterances which conform to a single canon of rules.
Both written and spoken utterances may be either formal or informal; but only a very practised orator can manage either the intricacies of fully formal speech or the vigor and effectiveness of good informal speech on the fly; and there are today very few of such speakers. So when the President or any other public figure gives a speech, even a fairly informal one, it is almost always written and read more or less verbatim. (It may appear to be spontaneous if the speaker is reading from a teleprompter.) Such addresses are followed too closely, and the consequences of mis-speaking are too weighty, to leave the form of expression to the speaker's improvisational abilities.