How to prove that these functions do not intersect?

Since you said that both functions are greater than one and strictly increasing, they vary so fast that I should instead consider $$F(x)=log(f(x)) qquad text{and} qquad G(x)=log(g(x))$$ If you plot them, you should notice that, as soon as $x > 3$, $F(x)$ and $G(x)$ looks very linear with $F(x) > G(x)$; I agree that this does not prove anything.

Now, consider a Taylor expansion of $F(x)-G(x)$ around $x=1$. This gives $$H(x)=F(x)-G(x)=(x-1)^2 left((pi ^2+1)+left(pi ^2-1right) cosh (2 pi )right)+Oleft((x-1)^3right)$$ So, starting from $0$ at $x=1$, $H(x)$ is an increasing function, which, according to Murphy's principle, will go through a maximum value. This maximum is located very close to $frac 98$ and its nature is confirmed by the second derivative test.

For sure, now the problem is : what happens for infinitely large values of $x$ ?

For the asymptotics analysis, I shall assume that, for large $t$, $cosh(t) sim frac 12 e^t$. Factoring the exponentials and ignoring the terms which are divided by $e^t$, we end with

$$f(x) sim frac 12 e^{2pi x} quad text{and} quad g(x) sim frac 12 e^{2pi x} frac{left(x^2+1right)^2}{left(x^2-1right)^2}implies H(x) sim frac{4}{x^2}+Oleft(frac{1}{x^6}right)$$ In other words, when $x$ is large $$f(x) sim g(x) , exp left({frac{4}{x^2}}right)$$

Computed exactly for $x=10$, the result is $H(10)=0.0400013$ !!

Edit

Looking at the last result, I computed as few values $$left( begin{array}{cc} x & x^2, H(x) \ 2 & 4.0859321 \ 3 & 4.0165826 \ 4 & 4.0052206 \ 5 & 4.0021354 \ 6 & 4.0010293 \ 7 & 4.0005555 \ 8 & 4.0003256 \ 9 & 4.0002032 \ 10 & 4.0001333 \ 11 & 4.0000911 \ 12 & 4.0000643 \ 13 & 4.0000467 \ 14 & 4.0000347 \ 15 & 4.0000263 \ 16 & 4.0000203 \ 17 & 4.0000160 \ 18 & 4.0000127 \ 19 & 4.0000102 \ 20 & 4.0000083 end{array} right)$$