Two different formulas for standard error of difference between two means I mostly see this formula when searching for a formula for the estimate of the standard error in difference between two means, and it is also used in this video.
$$\Delta=\sqrt{s_1^2/N_1+s_2^2/N_2}$$
But I've also seen this one (and this is the one my book uses):
$$\Delta'=\sqrt{\dfrac{\left(N_1-1\right)s_1^2+\left(N_2-1\right)s_2^2}{N_1+N_2-2}\left(\dfrac{1}{N_1}+\dfrac{1}{N_2}\right)}$$
As these are two very different formulas, how come they are used seemingly interchangeably?

In both scenarios $\sigma_{1}$ and $\sigma_{2}$ are unknown.  The bottom formula is using the assumption that $\sigma_{1} = \sigma_{2}$ and attempting to estimate that shared variance by pooling all observations together and calculating a weighted mean.  Thus, the factor on the left plays the role of both $s_{1}^{2}$ and $s_{2}^{2}$ in the bottom equation.  This method is usually used when you have small sample sizes and the equal variance assumption is plausible.
