I have found in one learning lesson that there is a difference between two words mentioned in title. But I can not find any audio evidence of this. Is there a real difference or it can be neglected in everyday use?
Both words are pronounced the same (in standard German): [ˈzaɪ̯tn̩], singular [ˈzaɪ̯tə]; source: Duden-Aussprachewörterbuch (3rd ed., 1990).
German orthography has a tendency to separate homophones wherever possible; similar cases are Leib/Laib, Lärche/Lerche.
Pronunciation is regionally different in Germany. In Swabia this example is pronounced like follows:
Saite [aɪ̯] or [ɔɪ̯] as in "Saitewürstle"
Seite [ɛɪ̯] as in "Gang uff'd Seite, I mecht vorbei", or "Seitenbacher"
Note however that this is inconsistent. Depending on the word, the Swabian pronunciation of "ei" may be [aɪ̯], [ɔɪ̯] or [ɛɪ̯].
The standard "hochdeutsche" pronunciation of Seite (like Saite) is considered as stilted in Swabia. Swabians therefore pronounce these words consistenly with their dialectal tone even when they do not speak dialect otherwise.
This may lead to confusion on how to appropriately pronounce when listening to natives having different regional roots.