There is a translation parallel to dejar de that is more common in British English than American: "(to) leave off". It means that something ongoing has ended or needs to stop.
So potentially the most literal and correct translation of this phrase would be that "It seems to me that it left off raining". I haven't tried it, and don't recommend it, but I will just put it out there as a way to remember the meaning of the Spanish.
They can both mean "it looks as if." Appear can also mean "to become visible." The word "seem" would never be used in that sense.
I would also say that when there is a difference, "seems" is probably just a little less certain than "appears." If someone says "that appears to be true," it can often mean something like "it's rather clearly evident that it's true." In that sense, it implies that there's an obvious conclusion. On the other hand, "seems" in the same sentence tends to imply that you believe something to be true even though all the facts may yet not be in.
Of course, as with all language, context and even the intonation of the speaker can completely change the meaning of what's said. If someone says "yeah, right," they mean neither "yes" nor "correct."
are=ser (to be) Definitely it has stopped raining Seems=me parece Seems is similar to "it appears, to be" It leaves room for error. In the context of this sentence "It seems to me that it stopped raining", one is not quite sure if the rain has stopped or slowed down temporarily.
Adding my two cents much later.
In addition to differences others have noted, seems emphasizes the subject and appears emphasizes the object.
"The object appears blue." I would understand this as describing a quality about the object, that the object itself looks blue.
"The object seems blue." I would understand this as describing a quality about the subject's perception, that it looks blue to the viewer(s).
Although either can be made to emphasize the other by some syntactic strategy ("It appears to me..."; "It itself seems...")
Gerunds are not used as often in Spanish as they are in English.
Frequently, the infinitive verb will play the role of a gerund in Spanish.
Here's a website that lists some of the uses of the infinitive that functions as a gerund: http://spanish.about.com/od/infinitives/a/infinitive_noun.htm
More links at the bottom of the page that might be useful too
I find it interesting that "llover" can be conjugated into all persons, in contrast with "pleuvoir" in French, which can only be conjugated in the third person singular. There are several other verbs like this in French, such as "falloir = to be necessary" and "traire = to milk a cow".
Can anyone who speaks Spanish (and perhaps French) comment on this? I was supprised to find that one can conjugate "llover" as "lluevo = I rain", which doesn't really make sense in English, as only the sky/ the clouds can rain.
According to my Spanish materials/references, "llover" is conjugated in third person singular only (just like "pleuvoir" in French).
llueve - il pleut - it rains
está lloviendo - il pleut - it is raining
Since French does not use a present continuous in the same way as Spanish or English, "il pleut" can mean "it rains" (e.g. "il pleut beaucoup en Angleterre" ("it rains a lot in England")) as well as "it is raining" (e.g. " maintenant, je ne veux pas sortir parce qu'il pleut" ("now I don't want to go out because it is raining")).
Where did you see "lluevo"?
To appear is "aparecer" and it does not share it's meaning with "parecer" (to seem) like it does in English
So when you use the word appear as a synonym for "seem" you use parecer
Apparently no, it should not. "As an opinion and understanding verb, parecer in Spanish needs subjunctive only when the verb is negative." (from http://spanology.com/spanish-lessons-verb-parecer-in-spanish/)
Me parece + indicative No me parece que + subjunctive
Strict English grammarians would say that you need "that" to join the two clauses together ("it seems to me" is the first clause; "it stopped raining" is the second clause). "That" as used here is a conjunction.
However, it is quite common for native English speakers to drop "that". You aren't wrong, just perhaps more casual. When I write, I sometimes leave "that" out, but if I think that the sentence isn't clearly indicating that there are two clauses, I will put "that" in. For me, it's more about clarity in expressing.
In this case, the "que" in Spanish matches the English "that". They are both used as conjunctions. For learning purposes, leaving "that" in the English translation will help English speakers understand why "que" is in the Spanish.
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It stopped = dejó de (preterite)
It has stopped = ha dejado de (present perfect)
Adding the has changes it to the present perfect tense with dejado in the past participle form, instead of conjugated in the preterite.
Duo is really picky about this sort of thing. I agree that "It seems to me that it has stopped raining" is a more natural sounding translation, but those are not the words that they gave us to translate, they gave us "it stopped"