Question

What is the difference between the two? And when should I use each of them?

Answer 1

There is none.

string.ToLower calls TextInfo.ToLower behind the scenes.

From String.cs:

    // Creates a copy of this string in lower case. 
    public String ToLower() {
        return this.ToLower(CultureInfo.CurrentCulture); 
    }

    // Creates a copy of this string in lower case.  The culture is set by culture.
    public String ToLower(CultureInfo culture) { 
        if (culture==null) {
            throw new ArgumentNullException("culture"); 
        } 
        return culture.TextInfo.ToLower(this);
    }