This discussion assumes friction is negligible. Friction, in fact, also exerts a force on the cart in a direction opposite to the cart's motion. Since the force from the string is greater than the force due to friction, there is a net force acting on the cart and it still accelerates..
If there is an acceleration, then the velocity is changing. This means that more and more distance will be covered for equal time intervals. Thus when x is plotted vs. t, the data points should follow a curve.
If the cart had been pushed (and released), the only force acting on the cart would be friction. These carts have pretty good bearings, so if the cart is pushed such that it is moving with a speed that is comparable to its final speed when being pulled by the weights, it should not slow down very much once released. That is, the cart should be moving with a fairly constant speed (compared to the first case). Thus equal distances should be covered in equal time intervals, and the plot of x vs. t should be much more straight than the first plot.