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We present results of a new deep, high-resolution, long-slit spectroscopic study of the extended nebular line emission around the young binary T Tauri. The deduced position-velocity diagrams of the [O I], [N II], and [S II] lines have been used to investigate in great detail the complex geometric and kinematic structure of the mass outflows from the stellar binary components T Tau N and T Tau S. A large number of distinct extended outflow components, each of them characterized by its specific spatio-kinematic properties, has been identified. In addition to the brighter components (A, B, C, D, E, and F) already described in detail by Böhm & Solf, we have detected several much fainter components (G, H, I, J, K, and L), some of them extending up to ∼40" from the central binary. The new data confirm the existence of two separate bipolar outflow systems associated with the binary: one of them oriented near the east-west direction and probably originating from the visible binary component T Tau N, the other one oriented near the north-south direction and probably originating from the infrared binary component T Tau S. In the east-west outflow, so far considered to be represented only by blueshifted components (B, F, and H) west of T Tau N, we have detected for the first time an indication for a redshifted faint counterjet (G) east of T Tau, corresponding to the known HH 155 jet (H). In the north-south outflow, so far considered to be represented only by two components (C and D) near the central source (≤3"), two additional, rather faint components (I and J) have been detected, which are located farther out (8"-15") on opposite sides of the source. Both faint components present rather low velocities of opposite sign but rather large velocity dispersions of about 75-100 km.s^–1 (FWHM). Using a biconical model for the outflow geometry, we have derived an inclination angle of 79° with respect to the line of sight for the bipolar axis and deduced typical velocities of ∼91 and ∼280 km.s^–1 in the faint outer regions (I and J) and the bright inner regions (C and D) of the north-south outflow, respectively. We have compared the kinematic state (mean radial velocity and velocity dispersion) of the ``hot'' (partially) ionized gas of the outflows from the binary T Tau deduced from our forbidden line observations with that of the ``warm'' molecular gas determined from the v=1-0 S(1) line of H₂ observations by T. M. Herbst and coworkers. In principal, the [S II] and H₂ emission regions refer to very different spatial regions of the outflows, and hence the kinematic states of the two regions do not have to be coupled to each other. In particular, it is expected that the velocity dispersion derived from the [S II] lines is much larger than that from the H₂ lines. These expectations are generally fulfilled by the observations, except for the ``core'' region of Burnham's nebula (component E) where the velocity dispersions of both the forbidden and H₂ lines are approximately the same. These findings as well as other enigmatic properties of Burnham's nebula reported earlier are not yet understood.