The "precision" of both methods is "double". You mean "accuracy". More details about the difference of these terms can be found at WikiPedia.
You cannot compare the accuracy of ODE15S and ODE45 directly, because the first method is appropriate for stiff systems, but the 2nd for non-stiff ones.
An analytically obtained Jacobian is not necessarily more accurate or faster than a dynamically determination using a numerical differentiation. A small mechanical system of e.g. 6 rigid bodies and some joints can lead to a enormous number of nested trigonometric functions in the Jacobian. Cancellation errors can amplify small numerical errors. When this leads to high frequent oscillations of the system, the step size control increases the number of steps, such that the the integration can take more time to produce worse results.
