What is the difference between an exponential func. and a hyperbolic secant func.?

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All coefficients/constants remaining the same, what is the dif. between them? physically and mathematically?

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Hyperbolic trig functions are built from combinations of exponential functions. I would say a defining characteristic on an exponential function call it $f$ is that $$f(x+y)=f(x)f(y)$$ i.e. $$e^{x+y}=e^xe^y$$ You can check that this is not the case for the hyperbolic secant function. $$sech(x)=\frac{2}{e^x+e^{-x}}$$ Another difference is that sech(x) is bounded; the limit at $\infty$ and $-\infty$ is finite.