Should you write \begin{equation} \Gamma(z) := \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0, \tag{1} \end{equation} or \begin{equation} \Gamma(z) = \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0?\tag{2} \end{equation}
I thought that $\Gamma(z)$ is defined by the integral $$\int_{0}^\infty e^{-t}t^{z-1}\,\,dt,$$
So $(1)$ is the correct way to introduce the Gamma function in a paper?
Are these differences important when writing a mathematical paper, or subjective and down to personal preference?
It is subjective.
My take is that if you are just reminding the reader of what your notation is, in conjunction with several other well-known facts, the $=$ sign is right. If you are, however, trying to make some subtle point that hangs on what is a definition and what is merely a fact, then the $:=$ is justified.