Are the words "Aural" and "Oral" usually pronounced the same? Does it vary by dialect? Are there strategies that people use to differentiate them when listening to spoken English?
Since this question asks about whether people can tell the difference I am going to answer for myself and my dialect rather than referring to a dictionary.
I speak Standard Southern English English and I pronounce them the same as far as I know. I cannot tell the difference when I hear them whatever accent the speaker has.
There is one exception and that is that I sometimes hear aural with the same vowel as now. Sometimes it is clear they are deliberately pronouncing it that way to distinguish it but sometimes I am not sure if they know how to pronounce it.
I never hear them pronounced slightly differently.
They are “supposed” to be pronounced the same in almost all accents of English (the exception would be accents without the “north-force merger”). It’s not unheard of for people to pronounce them differently, but none of the possible ways of differentiating them has become recognized as standard: they all involve pronunciations that many would consider “mispronunciations”. And I don’t know of any dictionaries that mention distinct pronunciations for these two words.
Aural has the long north vowel as the outcome of the thought vowel before /r/
Words with "au" followed by /r/ are not that common, so it's a little hard to describe their pronunciations in terms of a general rule. Dinosaur, Minotaur, centaur, thesaurus, tyrannosaurus are other words where "aur" is standardly pronounced as the vowel found in north.
Laurel has a pronunciation with the "short o" vowel in British English, /ˈlɒrəl/, which can be considered irregular. (A few words spelled with au before another consonant also have irregular pronunciations with "short o" in British English, such as sausage /ˈsɒsɪdʒ/.)
As mentioned in David Robinson’s answer, aural is sometimes pronounced with the sour diphthong. The phonetician John Wells made a blog post mentioning the "ˈaʊrəl" pronunciation of aural, and the comments below it have some more discussion: "trauma", John Wells's Phonetic Blog.
From an American English speaker with the cot-caught merger, “aural” might be heard with the unrounded back vowel of “caught” (vs. a rounded vowel in “oral”). For anecdotal evidence, see the posts here: http://www.verduria.org/viewtopic.php?f=4&t=73&start=200 as well the comments left below your question by Hot Licks and choster.
Oral has the long force vowel as the outcome of the "long o" vowel before /r/
Oral is an -al adjective, a class of words that come from Latin and that generally are pronounced in a way consistent with the "traditional English pronunciation of Latin words". In the case of oral, the relevant part of the traditional Latin pronunciation rules is a rule saying that a vowel letter represents a "long vowel" when it comes in the second-to-last syllable of a word and is followed by only a single consonant letter. Since the O in "oral" meets these criteria, it is regular for it to be pronounced as a "long o" sound, which before a /r/ is normally modified into the force vowel.
There are exceptions to the traditional pronunciation rules: the most relevant one is moral, which British English speakers pronounce as /ˈmɒrəl/, with a "short o" sound. (Coral also has a "short o" in British English pronunciation, but it is not an -al adjective.) But oral doesn't seem to be a well-established exception. The Teflpedia article "Pronunciation exercises: /ɒrV/ vs /ɔːrV/" points out that the Collins Dictionary entry for "oral in British" lists the variant pronunciation /ˈɒrəl/ after /ˈɔːrəl/.
I have always lived in Southern England, and am a bit of a language pedant. I don’t know if others would consider it correct, but I always distinguish by pronouncing aural like choral, and oral like coral. I hate it that so many pronounce them both like choral. And pronouncing aural like “owral” makes it sound like a German word.