I am looking for a pronunciation of the word слезы (tears) and saw on forvo.com both spellings слезы and слёзы which are pronounced differently. Is the meaning the same in both cases and how/why do they differ?
They are pronounced exactly the same way because it is exactly same word. The usage letter "ё" from the very beginning of its existence (it was introduced at the end of XVIII century) was quite controversial and almost never - obligatory.
That leads to this double spellings. In some cases it finally affected the way we pronounce some words (nobody's saying Рёрих nowadays, other very famous example is "свекла"/"cвёкла"), but this never happened with "слёзы", which obeys regular rules of pronouncing words with "ё" ("йо" at the beginning, palatalization of preceding consonant plus "о" anywhere else).
Keep in mind though, that singular form, "слеза" is exactlу without "ё", the same, for instance, with "звезда"/"звёзды".