In the specific case, calling First and Where on a string[], the methods called are the Enumerable.Where and Enumerable.Firstextension methods.
Enumerable.Where does this:
public static IEnumerable<TSource> Where<TSource>(this IEnumerable<TSource> source, Func<TSource, bool> predicate) 
{
  // null checks omitted
  if (source is TSource[]) 
     return new WhereArrayIterator<TSource>((TSource[])source, predicate); 
  //the rest of the method will not execute
}
and the constructor of WhereArrayIterator does just:
public WhereArrayIterator(TSource[] source, Func<TSource, bool> predicate) {
  this.source = source; 
  this.predicate = predicate;
} 
So nothing is actually done here, except to create an iterator.
The first First method, without a predicate does this:
public static TSource First<TSource>(this IEnumerable<TSource> source) { 
  //null check
  IList<TSource> list = source as IList<TSource>;
  if (list != null) {
     //this branch is not taken as string[] does not implement IList<string>
     if (list.Count > 0) return list[0]; 
  }
  else { 
    //this is actually the WhereArrayIterator from before
    using (IEnumerator<TSource> e = source.GetEnumerator()) { 
      if (e.MoveNext()) 
        return e.Current;
    } 
  }
  throw Error.NoElements();
}
However, the second First does this
public static TSource First<TSource>(this IEnumerable<TSource> source, Func<TSource, bool> predicate) {
   //null checks
   foreach (TSource element in source) {
     if (predicate(element)) return element; 
   }
   throw Error.NoMatch();
}
Which in the case of an array, is as fast as direct linear access.
In a nutshell, this means that calling First(predicate) on an array will be somewhat faster, by a not large, but still detectable factor. This might not hold for lists, and will certainly not hold for IQueryable objects, that are a completely different story.
However, this is micro-optimization at it's worst. Unless this is done millions of times, it won't save too many seconds. Even as I know this now, I'll still use whatever is clearer to read and understand.