Among current and voltage which is responsible for brightness of a bulb? In a circuit that is fitted with a bulb, which is the factor that affects brightness of that bulb: current in the circuit or the voltage offered by the battery in that circuit?
 A: which is the factor that affects brightness of that bulb, current in the circuit or the voltage
Both.  It's the power that ultimately causes the filament to get hot and emit visible black body radiation.
Power is voltage times current, so both matter.
However, you can only control one degree of freedom.  The bulb dictates the other.  This single degree of freedom can be expressed various ways.  Two of them are fixing the current and fixing the voltage.  Once you fix one of these, the resistance of the bulb implicitly fixes the other.
Note that the resistance of a bulb varies considerably with temperature.  It is much higher when the bulb is emitting light than when it is sitting cold and unpowered.  However, that still doesn't let you fix both independently.  It only means that the relationship between voltage and current changes with the set point.
A: If you think of the lighting element, which may be a length of tungsten or the like, as a fixed resistor of R ohms, the current in the resistor is determined by:
$$ v = R~ i, $$
Ohm's Law.  The higher the voltage v the higher the current.  If the brightness is caused by a flow of electrons through the filament, a higher voltage will--all things being equal-- drive more electrons to flow through the filament. 
So the brightness is a function of both current and voltage, and can be said to depend on both. That is, we can write $v(i) = R~i$ or $i(v)=v/R.$
We might be tempted to say that the intensity I of the light is proportional to current, $I \propto i $ but we would have to remember that $i $ is a function of $v$ and put $I \propto i(v)=v/R.$
This can get arbitrarily complicated, since intensity of the light may not be proportional to current in a simple way, and the precise behavior of electrons in a circuit depends on the exact conditions and the nature of the components. For example, as comments below note, the resistance may not be constant, but may be a function of temperature. 
An accessible article on how resistance may vary in a light bulb is given here. A plot of I vs. V shows that the resistance R is not in general constant. To characterize the circuit in terms of V and I you might have to solve $\frac{dv}{di}=f(i)$ or take some measurements to get a quantitative idea.
