We report a systematic study of the superconducting (SC) and normal-state anisotropy of Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ single crystals with controlled amounts of excess Fe ($y$ = 0, 0.07, and 0.14). The SC state anisotropy $gamma_{H}$ was obtained by measuring the upper critical fields under high magnetic fields over 50 T for both $Hparallel ab$ and $Hparallel c$. On the other hand, the normal state anisotropy $gamma_{rho}$ was obtained by measuring the resistivity with current flowing in the $ab$ plane ($rho_{ab}$) and along the $c$ axis ($rho_c$). To precisely measure $rho_{ab}$ and $rho_c$ in the same part of a specimen avoiding the variation dependent on pieces or parts, we adopt a new method using a micro-fabricated bridge with an additional neck part along $c$ axis. The $gamma_{H}$ decreases from a value dependent on the amount of excess Fe at $T_{rm{c}}$ to a common value $sim$ 1 at 2 K. The different $gamma_{H}$ at $T_{rm{c}}$ ($sim$1.5 for $y$ = 0, and 2.5 for $y$ = 0.14) suggests that the anisotropy of effective mass $m_c^*/m_{ab}^*$ increases from $sim$ 2.25 ($y$ = 0) to 6.25 ($y$ = 0.14) with the excess Fe. The almost isotropic $gamma_{H}$ at low temperatures is due to the strong spin paramagnetic effect at $Hparallel ab$. By contrast, the $gamma_{rho}$ shows a much larger value of $sim$ 17 ($y$ = 0) to $sim$ 50 ($y$ = 0.14) at the temperature just above $T_{rm{c}}$. Combined the results of $gamma_{H}$ and $gamma_{rho}$ near $T_{rm{c}}$, we found out that the discrepant anisotropies between the SC and normal states originates from a large anisotropy of scattering time $tau_{ab}$/$tau_c$ $sim$ 7.8. The $tau_{ab}$/$tau_c$ is found to be independent of the excess Fe.
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