To be more precise the \$h_{ie}\$ is not strictly equivalent to \$(\beta+1)r_e\$.
\$h_{ie}\$ also known as \$h_{11}\$ comes directly from a two-port network theory. Where they treat the transistor as a black box. http://ux.brookdalecc.edu/fac/engtech/andy/engi242/bjt_models.pdf
On the other hand, the \$r_e\$ is coming directly from the Shockley equations.
But despite this in hand calculation, we are using this approximation
\$h_{ie} \approx r_\pi = (\beta+1)r_e\$ in small signal analysis (when BJT working in linear/active region).
But BJT also has some "real" resistances "built into" the BJT's terminals.
The base spreading resistance (\$r_{bb}\$)
And "bulk" emitter resistance around \$0.5\Omega\$ for small-signal BJT's.