I am really confused which one is more accurate... Even the pronunciations are different.
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Both of the sources that you linked to indicate variation. Merriam-Webster says "Sassanian [...] variants: or Sasanian". Wikipedia says "The Sasanian Empire [...] also known as the Sassanian, Sasanid, Sassanid or Neo-Persian Empire".
The name of the dynasty/empire is considered to be related to the personal name that Wikipedia writes as Sasan/Sāssān.
I don't know the explanation for the variation between -s- and -ss- here. We do see such variation in some other English words; for example, disyllabic has a (now rare) spelling variant dissyllabic, in which the (unetymological) use of -ss- seems to have been motivated by the fact that the word is pronounced with a voiceless /s/ rather than voiced /z/.
Based on the Google Ngram Viewer, the spelling "Sassianian" seems to have originally been dominant (the word itself seems to have started being used in the second half of the 18th century), with the spelling "Sasanian" rising in frequency from around 1800 to eventually overtake "Sassanian" sometime around the 1970s or 80s. The Corpus of Contemporary American English (COCA) also shows greater frequency for "Sasanian" (27) than for "Sassanian" (8).
The etymology section of the Oxford English Dictionary entry for Sassanid seems to indicate that -ss- spellings of related words existed in medieval Latin: supposedly, Sassanid comes from "medieval Latin Sassanidae plural, < Sassan, Sasan". I haven't been able to find out much more about these Latin words yet, but if this is accurate, then the use of "ss" spellings for this name (and related words) did not originate in English. English names for figures from antiquity often are based on Latin or Greek transcriptions, which were not always systematic or regular. An answer post by varro on the Latin SE site suggests that Greek -σσ- (rendered in Latin and English transliteration as -ss-) may have been used in the transcription of non-Greek certain names to represent some unclear difference between sibilants in foreign words and the sibilant /s/ sound found in native Greek words. It seems the use of -ss- in the modern English spelling of the word "Messiah" can be traced back to Greek. However, the Portuguese Wikipedia (an odd choice of source, I know! but it was the only one I could find with a Google search) gives the Classical Greek spelling as Σασαν, so I don't know if Greek influence is actually behind the -ss- spellings of the name Sas(s)an. There do seem to be multiple hits for the spelling "Σασσαν" on Google search, but I don't know how to evaluate that information. There is a proposal for a Greek Language SE site, and questions about Ancient Greek are on-topic for the currently active Latin SE site; I would recommend following and posting questions on these sites if you have questions about the history of Greek spelling, pronunciation and transcription conventions.
Based on brief research that I have done since reading your question, my understanding is that double or lengthened consonants do exist/have existed in the pronunciation of at least some inherited Persian vocabulary, but were not necessarily spelled any differently from single consonants. You would need to ask an expert on the history of Persian to know if this name ever was pronounced with a long medial consonant in Persian; I can't figure this out (although it seems that, as in many languages, it is at least a bit unusual for a long consonant to occur after a long vowel in Persian). If a pronunciation with a long consonant existed in Persian, it might be transcribed in English with two consonant letters, even if the original spelling used one consonant letter. Compare the situation with the conventional English transcriptions of Arabic: the Arabic noun حج is only spelled with one letter jīm, but is conventionally transcribed in English as "Hajj" (though Wikipedia says the spelling "Haj" has also been used).
The pronunciation variants don't seem to have any connection to the spelling variations. Merriam-Webster suggests that the vowel in the first syllable (which is unstressed) may be pronounced as reduced /ə/ or as unreduced /æ/. Wikipedia suggests that the vowel in the stressed syllable may be pronounced as /eɪ/ or /ɑː/. The Oxford English Dictionary lists a third option, /æ/, although this is only given for American English.
The existence of this kind of variation can be explained in terms of different speakers applying different English pronunciation patterns to the word. For example, there is one pronunciation rule that says that the letter "a" is pronounced as /eɪ/ in this context (see my previous post Why is "salient" pronounced with a "long a" sound? for more details and background about that rule), but there is also a tendency to pronounce "a" as /ɑː/, or sometimes /æ/, in certain words from foreign languages. (I'm also reminded of an ELL question that I answered a little while ago that deals with similar variation in the pronunciation of the vowel in the stressed syllable of "Eulerian".) The application of vowel reduction in unstressed syllables varies substantially between accents and even between different speakers of the same accent; see tchrist's recent answer to the question How to pronounce -on endings? for some more examples of this.
Because there are conflicts between different patterns for English pronunciation, neither pronunciation in a context like this can be identified as definitively "correct".
If you meant to ask about Persian pronunciation, that's not really within the scope of this site, but my understanding is that [ɑ] is phonetically closer than [eɪ] to the value of the vowel that occurs in both syllables of the name Sāsān. Wikipedia transcribes the standard modern Iranian Persian pronunciation of ā as /ɒː/. But of course, in Persian as in English there are differences in vowel quality between accents, and there have been some changes between different periods of the language, so you would have to ask an expert in Persian to get a good understanding of the pronunciation(s) of this name.
There is no "correct" spelling of foreign names: see this question as an example of the discussion. This particular example is more extreme than most; Farsi is extremely difficult to map into English, and the pronunciation of both languages has changed over the years. What is more, several extinct languages, like Neo-Elamite and Scythian, played an important part in the Empire; nobody has any idea how those were pronounced (and so should be transliterated).
Those who have made an intensive study of the period are entitled to a view as to how best to spell the name (I have no doubt that some academic conferences are fiercely divided between single-s and double-s), but the rest of us can and do use either.
Both spellings are correct and official.
On the Ancient History Encyclopedia
The Sasanian Empire (also spelled Sassanian, Sasanid or Sassanid) was the last pre-Islamic Persian empire, established in 224 CE by Ardeshir I, son of Papak, descendant of Sasan.
(Alonso Constenla Cervantes, Sasanian Empire )
On the Encyclopædia Britannica Sāsānian dynasty - Iranian dynasty
Sāsānian dynasty, also spelled Sāssānian, also called Sāsānid, (ad 224–651), ancient Iranian dynasty evolved by Ardashīr I in years of conquest, ad 208–224, and destroyed by the Arabs during the years 637–651. The dynasty was named after Sāsān, an ancestor of Ardashīr I.
Classically it was spelled with a double-s although the alternate also existed in a small way. It's only in the sixties that the new spelling found a sudden spurt in use and use of the classical spelling declined rapidly. Sasanian is settling down as the spelling. However, since a large body of work created over the last couple of centuries could be existing, using the older spelling, it makes sense to recognize both spellings "for historical reasons."
(Ref: nGrams "Sassanian,Sasanian" -- unable to load image at this time.)