I have come across both of these terms when searching words via google. Is there a difference between these two terms, or is it just a case of one dictionary prefering one term over another?
According to dictionary.com,
The meaning of these temporal labels can be somewhat different among dictionaries and thesauri. The label archaic is used for words that were once common but are now rare. Archaic implies having the character or characteristics of a much earlier time. Obsolete indicates that a term is no longer in active use, except, for example, in literary quotation.
Thus, "obsolete" is somewhat of a stronger word than archaic, indicating that a word may be no longer in use at all, while an "archaic" word may be old-fashioned and rarely used but still understandable or occasionally used. Since the definition does vary, the best place to learn the distinction between these terms for a particular dictionary isn't from this site, but from the dictionary itself: dictionaries do generally have explanations of the terms they use in entries.
To follow up on sumelic's succinct answer, I note that Dictionary.com is defining archaic and obsolete specifically with respect to usage labels. Each dictionary is likely to have a different specific way of defining the temporal labels it uses in handling its word entries. For example, Merriam-Webster's Eleventh Collegiate Dictionary (2003) has this note:
Three types of status labels are used in this dictionary—temporal, regional, and stylistic—to signal that a word or sense of a word is not part of the standard vocabulary of English.
The temporal label obs for "obsolete" means that there is no evidence of use since 1755: [example omitted] The label obs is a comment on the word being defined. When a thing, as distinguished from the word used to designate it, is obsolete, appropriate orientation is usually given in the definition: [examples where "ancient" and 16th century" are included in the definition to indicate that the thing—as opposed to the word—is not now in use omitted].
The temporal label archaic means that a word or sense once in common use is found today only sporadically or in special contexts: [examples omitted].
The American Heritage Dictionary of the English Language, fourth edition (2000) offers remarkably similar comments about its temporal usage labels:
Temporal labels. Temporal labels signal words or senses whose use in modern English is uncommon.
Archaic. This label is applied to words and senses that were once common but now are rare, though they may be familiar because of their occurrence in certain contexts, such as the literature of an earlier time. Specifically, this label is attached to entry words and senses for which there is only sporadic evidence in print since 1755: [examples omitted].
Obsolete. The label Obsolete is used with entry words and senses no longer in active use, except, for example, in literary quotations. Specifically, this label is attached to entry words and senses for which there is little or no printed evidence since 1755. [Example omitted.]
But the Concise Oxford English Dictionary, revised tenth edition (2002), which refers to such labels as "register labels," doesn't use the word obsolete as a label at all. Instead these are the most relevant register labels that the Concise OED uses:
dated: no longer used by the majority of English speakers, but still encountered, especially among the older generation.
archaic: old-fashioned language, not in ordinary use today, though sometimes used to give a deliberately old-fashioned effect and also encountered in the literature of the past.
historical: still used today, but only in reference to some practice or artefact that is no longer part of the modern world, e.g. banneret or umbo.
poetic/literary: found only or mainly in poetry, or in literature written in a consciously 'literary' style.
rare: not in normal use.
And notwithstanding its name, the Longman Dictionary of Contemporary English, new edition (1992) includes several "labels showing time," though neither archaic nor obsolete is among them:
Labels showing time
Some words are no longer used in modern English (though thy will often be found in old books), and some are beginning to be used less often. These are shown by the following labels:
old-fash old-fashioned — no longer common, used mainly by older people
old use no longer used
rare or becoming rare raely used or beginning to be used less often
Finally, some dictionaries don't explain the precise meaning of their labels at all. For example, The Random House College Dictionary (1984) includes Archaic and Obs. among it "restrictive labels," but it doesn't directly address how they differ. To figure that out, you have to go to the relevant definitions in the body of the dictionary:
archaic adj. ... 2. (of a linguistic form) current in an earlier time but rare in present usage.
obsolete adj. ... 3. (of a word or other linguistic unit) no longer in use, esp., out of use for at least a century.
The upshot of these different labeling schemes is that no fixed, universally recognized meaning of archaic and obsolete exists, even in the narrow sense of temporal (or register) usage labels in dictionaries. It does seem generally true that something obsolete is farther gone than something archaic is; but it would be a serious mistake to suppose that historical use or nonuse in 1755 of a word that is no longer used today conclusively determines whether that word is obsolete or archaic—unless you accept the Merriam-Webster/American Heritage labeling conventions.
Archaic words are those which are still used in literary sense of meaning like in Poems, Novels, or to add more attention on a sentence.
Whilst, obsolete words are those which have been discontinued from the usage, and are way too awkward or impossible to interpret.