I know the following two definitions for the delta function and Kronecker delta, respectively:
(1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$
(2) $\int_{-\pi}^{\pi}\frac{e^{i(m-n)t}}{2\pi}\mathrm{d}t=\delta_{m,n}$
First question: What is the difference between the delta function and the Kronecker delta? Especially in view of eq. (1), couldn't I say (2) $=\delta(m-n)$? Or would this mean something different?
Second question: Why are the integration limits different? I would think that I could replace $w$ in (1) by $(m-n)$ to get to definition (2), but obviously this wouldn't work because of the different integration limits.