The likeliest problem vs. the most likely problem:
- are they both correct?
- do they mean the same thing?
- is one preferable over another?
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They're both equally understandable and clear, they both mean the same thing, and they're both grammatically correct, but one is stylistically better (and, therefore, preferable). This is a style choice (personal judgment and preference prevail, in other words). My style is generally formal, so I prefer "the most likely problem"; "the likeliest problem" is casual and more appropriate for spoken English or for a letter to a friend. Superlatives with two-syllable words such as "likely" usually give the writer/speaker the choice of adding the /-est/ suffix or using "the most..." structure. One wouldn't say "the most big" (one-syllable word) but "the biggest", and one wouldn't say "the interestingest" (three-syllable word) but "the most interesting".
At first, comparison of disyllabic adjectives is a very tricky thing - cf. Palmer, Huddleston, and Pullum 2002: 1584 "there is no hard and fast boundary between those that can inflect and those that can't: speaker judgements are by no means wholly uniform." A good rule of thumb for disyllabic adjectives is that the analytic forms (with more and most) are always possible, unlike the inflectional ones (with -er and -est).
However, adjectives ending in "-ly" generally permit inflection (PHP 2002: 1583). The authors of the "Longman Grammar of Spoken and Written English" (LGSWE) make a very important addition to that statement, "with varying degrees of frequency" (Biber et al. 1999: 523). Quite often it's a matter of style/register (Weiner and Delahunty 1994).
To answer your specific question, Biber et al. argue that with the adjective "likely," the analytic forms ("more likely") are much more common that the inflectional ones ("likelier"), esp. in academic prose and in news, where you're more likely to encounter the analytic forms in general.