Rats make rather abrupt movements too. I would expect the difference you observe in movement speed is caused by size more than it is caused by being a bird vs a mammal. But why size would matter?
An increase in the linear dimension $x$ lead to an increase in the cross-section of muscles $x^2$ and an increase in the volume (and hence mass) of $x^3$. The strength of muscles is affected by their cross section $x^2$. Hence as animal grow bigger, the ratio $\frac{x^2}{x^3} = \frac{1}{x}$ becomes smaller and so does their ability to create large acceleration of their body parts. Their movement become smoother and less abrupt.
You might want to have a look at this highly related post too.