The distinctions between $\Lambda$CDM and $f(T)$ gravity according
  Noether symmetry
  Noether's theory offers us a useful tool to research the conserved quantities
and symmetries of the modified gravity theories, among which the $f(T)$ theory,
a generally modified teleparallel gravity, has been proposed to account for the
dark energy phenomena. By the Noether symmetry approach, we investigate the
power-law, exponential and polynomial forms of $f(T)$ theories. All forms of
$f(T)$ concerned in this work possess the time translational symmetry, which is
related with energy condition or Hamilton constraint. In addition, we find out
that the performances of the power-law and exponential forms are not pleasing.
It is rational adding a linear term $T$ to $T^n$ as the most efficient
amendment to resemble the teleparallel gravity or General Relativity on small
scales, ie., the scale of the solar system. The corresponding Noether symmetry
indicates that only time translational symmetry remains. Through numerically
calculations and observational data-sets constraining, the optimal form $\alpha
T + \beta T^{-1}$ is obtained, whose cosmological solution resembles the
standard $\Lambda$CDM best with lightly reduced cosmic age which can be
alleviated by introducing another $T^m$ term. More important is that we find
the significant differences between $\Lambda$CDM and $f(T)$ gravity. The
$\Lambda$CDM model has also two additional symmetries and corresponding
positive conserved quantities, except the two negative conserved quantities.
