Why are there two formats for the Solow capital evolution equations? 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?
The Solow-Swan has the two versions, one is in discrete time (first equation) and other in continuous time. Both give the same results, but you cannot mix them - as you did. 

In the first equation (discrete time) you can take the first diference but never a derivative, since the variable is not continuous. The point above the variable means that it has been diferenciated with relation to time.

So the last equation does not make any sense because tried to mix both models.