Wiener Process

[Jcrew]

So a Wiener process can be modeled by the following difference equation:

W(t) = W(t-1) + N(0,1)

where N(0,1) = normal distribution with mean 0 and unit variance 1.

Somewhere I saw someone use the following closed form for it:

W(t)= sqrt(t) * N(0,1)

Is the closed form an approximation for the difference equation? If not can someone show the steps to reach to the close form from the difference equation?