Are there any difference between the following sentences in meaning?
I met a beautiful girl.
In comparison to:
I met a girl who is beautiful.
Are the both grammatically correct and which one is better in writing?
As @krobelusmeetsyndra has correctly noted:
The one additional point I would like to make is that the longer construction is useful if you want to refer to the girl's other qualities. For instance, if the girl were also intelligent and strong, then the longer of the two options below would sound better (although, again, both are correct).
I met a beautiful, strong, intelligent girl.
I met a girl who is beautiful, strong and intelligent.
Again, though, these are just aesthetic differences, as both are correct.