/ˈaɪən $ ˈaɪən, ˈaɪɑːn/
UK /ˈaɪ.ɒn/ US /ˈaɪ.ɑːn/
Why does the pronunciation of these two words differ in these two sites?
It's a partial difference between American and British English. In American English, only /ˈaɪ.ɑːn/ is correct. In British English, it appears that both /ˈaɪ.ɑːn/ and /ˈaɪən/ are acceptable.
/ˈaɪ.ɑːn/ may be more prevalent or technically correct in the UK, and so Cambridge decided to only use it, though Longman and other sources (such as Wiktionary) probably decided to include pronunciations that many people use, even if it's less correct.
Of the two, since /ˈaɪ.ɑ(ː)n/ is correct in both UK and US English, I would recommend using that pronunciation.
(It's also worth noting that the difference between /ɑː/, /ɑ/, and /ɒ/ is generally non-existent among native speakers, especially in non-stressed syllables. They all represent the same phoneme, even if the realization is slightly different.)