Seem = Suggests how something looks or seems to look; used to describe a perceived condition; also suggests that something is true when we are not certain or when we want to be polite.
Appear = Used to talk about facts and events describing an observable condition; denotes that the opinion is based on a general visual impression. It also suggests that we do not quite believe that somebody or something really is as they seem:
His car seems/ appears to have stuck in a traffic jam.
It seems a best bet to take a taxi to the airport.
There seems to be some confusion over who is actually presiding over the function.
It seems crazy that we should do it all over again. √
It appears crazy that we should do it all over again. ✗
It seems like she’ll never come around.
It seems as if they’ve parted their ways following their disagreement about the company’s title.
They seem to have lost their way; they’d better take help of Google maps?
The car seems to be in good condition, so worth its price.
The major difference between ‘seem’ and ‘appear’ is the ‘certainty-uncertainty’ effect. With ‘seem’, the observer exhibits the uncertainty while with ‘appear’, the uncertainty is exhibited by the attributes of the observed person or thing which can also be deceptive:
She appears very upset – she has got some bad news.
There appears to be some problem with the ATM; I can’t get the money out.
The minister doesn’t appear to know about the number of women in the Civil Service.
Things did not go as they appeared – everything fell flat.
He didn’t appear so smart but he fixed the leaking tap in no time.
I didn’t want to appear rude – it’s just that there was no room in the car.
Is she as worried about Martha as she appears?
In the beginning, everything appeared normal.
The prisoners appeared (to be) rather contented with what they get.
She appears to have run off with his friend and can’t be traced now.
So, we can conclude that ‘appear’ is less common than ‘seem’ and ‘appear’ is a bit more formal too.
Appear + adjective + noun:
He appears quite a nervous candidate for the interview. (less often; in more formal situations)
It appears + as if/ as though/ that:
It appears as if Mr. Wilson hasn’t got his facts straight.
It appears as though the police haven’t verified the facts before arresting him.
It appears that she hasn’t been called for the audition.
He likes to appear as if he knows everything about the automobiles, but in reality he was only pretending.
It appears that none of the occupants of the car were injured.
It would appear (that) (= it seems that) no authorities have granted permission for the demonstration to take place.
She was extremely angry but now appears calm.
Though in the beginning he appeared (to be) rather stranger, later on we exchanged friendly greetings.
Appear + to infinitive:
He appears to hate me – even he’s causing others in the class to dislike me.
There appears to be some human error which caused the air crash.
It appears to me (that) he has no experience to manage large projects. (= I don’t think…)
It appears not/so:
“Are the kids busy with their homework?” “It appears not/so.”
“I think he has left the house by the back door.” “So it appears.”
‘Try to appear’ or ‘try to seem’:
Jack tried to appear calm when he was interrogated by the officers about who he was and what he had been doing. (Not tried to seem….)