Why is one of those spelled with a single L and not the other?
For the etymology of Beryllium name it's unclear but could be either Greek or Latin, and Helium is named after Helios (so Greek here).
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As the comments have said, there isn't any reason to expect these two different words to be spelled with the same number of Ls. The use of double consonant spellings in English is often related to either pronunciation or etymology.
The spelling of helium can be explained in terms of pronunciation. Helium has a "long e" sound in the first syllable. Double consonants are not usually used after vowel letters that represent "long" vowel sounds. "Hellium" would only be a plausible spelling for helium if the first syllable were pronounced with the vowel of hell rather than with the vowel of heel.
The spelling of beryllium can be explained in terms of etymology. As mentioned in the comments, both Greek βήρυλλος and the derived Latin beryllus have a double consonant in the same position.
Etymology also works as another way to explain the spelling of helium, since Greek ἥλιος has a single λ.
The spelling of beryllium can't be explained in terms of pronunciation alone. A word spelled berylium would likely be pronounced the same way: compare syndactylia, which has a "short i" sound in the third syllable even though it is spelled with a single L. The "short i" pronunciation of Y in this context follows a rule that I explain in my answer to the question Why do we pronounce a long second vowel in “decide”, but a short second vowel in “decision”?
In my answer to Adding an L when appending an -ium suffix to a word? (Metallium vs. Metalium), I give more examples of words that are spelled with double L for etymological reasons.
The answer to why Helium has one "L" and Beryllium has two is the number of syllables.
First lets deal with Helium. Hel is one syllable when we add ium we do not add an extra "L" However from which ever source you wish to take Beryllium whether it has one or two "L"'s it always has more than one syllable. Quote Some words, several of them ending in "L", with more than two syllables, have a double consonant even though the last syllable is not stressed;
Cambridge English dictionarySome words, several of them ending in l, with more than two syllables, have a double consonant even though the last syllable is not stressed; for
example, labelling, traveller, equalled, handicapped, programmed.
In American English the single consonant spelling is usually more common: labeling, traveler.
Beryllium Etymology wikipedia Early precursors of the word beryllium can be traced to many languages, including Latin beryllus; French béry; Ancient Greek βήρυλλος, bērullos, 'beryl'; Prakrit वॆरुलिय (veruliya); Pāli वेलुरिय (veḷuriya), भेलिरु (veḷiru) or भिलर् (viḷar) – "to become pale", in reference to the pale semiprecious gemstone beryl. The original source is probably the Sanskrit word वैडूर्य (vaidurya), which is of South Indian origin and could be related to the name of the modern city of Belur. For about 160 years, beryllium was also known as glucinum or glucinium (with the accompanying chemical symbol "Gl", or "G" ), the name coming from the Ancient Greek word for sweet: γλυκύς, due to the sweet taste of beryllium salts.
Helium wikipedia is named for the Greek Titan of the Sun, Helios. It was first detected as an unknown, yellow spectral line signature in sunlight, during a solar eclipse in 1868 by Georges Rayet, Captain C. T. Haig, Norman R. Pogson, and Lieutenant John Herschel, and was subsequently confirmed by French astronomer, Jules Janssen. Janssen is often jointly credited with detecting the element, along with Norman Lockyer. Janssen recorded the helium spectral line during the solar eclipse of 1868, while Lockyer observed it from Britain. Lockyer was the first to propose that the line was due to a new element, which he named.