Where is the difference between the union and sum of sets?

Take this example to clarify the difference: $$V=\mathbb{R}^{2}$$ $$W_{1}=sp_{\mathbb{R}}\{(1,0)\}=\{(a,0)|a\in\mathbb{R}\}$$ $$W_{2}=sp_{\mathbb{R}}\{(0,1)\}=\{(0,b)|b\in\mathbb{R}\}$$

Then,

$$W_{1}+W_{2}=\{w_{1}+w_{2}|w_{i}\in W_{i}\}=\{(a,0)+(0,b)|a,b\in\mathbb{R}\}=\{(a,b)|a,b\in\mathbb{R}\}$$

but,

$$W_{1}\cup W_{2}=\{v_{1}|v_{1}\in W_{1}\}\cup\{v_{2}|v_{2}\in W_{2}\}$$ and this set is consistent of all elements of the form $(a,0)$ and $(0,b)$ (where $a,b\in\mathbb{R})$ but, for example, $(1,1)\not\in W_{1}\cup W_{2}$.

$$W_1+W_2=\{w_1+w_2\mid w_1\in W_1\land w_2\in W_2\}\;.$$