I have seen the capital transition equation take two forms
$$K_{t+1} = sY_t +(1-\delta) K_t $$ $$\dot{K}_t = sY_t - \delta K_t$$
Obviously I can rewrite the top one to get
$$K_{t+1} - K_t = sY_t - \delta K_t $$
So that suggests somehow
$$\dot{K}_t \equiv K_{t+1}-K_t$$
But as far as I know,
$$\dot{K}_t \equiv \frac{\partial K_t}{\partial t} = \lim_{h\rightarrow 0} \frac{K_{t+h} - K_t}{h}$$
which is, as far as I know, not the same as $K_{t+1}-K_t$.
my question
Is $$\dot{K}_t \equiv K_{t+1}-K_t$$ right? If so, what does this mean and why isn't it the usual derivative?