round is British English, for example, "Let's go round the room and introduce ourselves." Americans would say "around the room" in this sentence. So in your example they're both correct and have the same meaning. One is British and the other is American. Of course "round" is also an adjective (the world is round, a circle is round, etc.) and you can't use "around" in a sentence like this, only "round."

'round' is an adjective or a noun. To compare it with 'around' it is enough to mention its meaning as an 'adjective'. You can't really say 'round the earth' ! 'round' means having a circular shape or object , maybe you can say ' a round earth' ,describing it ,but it isn't fully round so it is not an accurate description as well. 'round' means also ' complete or entire. It could also mean 'full or plump' as in 'round cheeks'. When speaking about speech you could describe it as straightforward or candid by using the adjective 'round' as in 'round assertion'. As for 'around' it could be a preposition that means to be situated near a point of reference in all directions ,here and there as in 'around the earth' . As an adverb 'around' means in the vicinity ,present in some place, where the exact location is not determined. Informally 'around' means also (as an adverb) someone who has been to many places, so as to have gained considerable experience, often of a worldly or social nature . "He has been around."

Generally speaking, there is no difference other than the spelling. Certainly not worth worrying about.