Difference between recursive B-spline algorithm and convolution

Is it true that the recursive algorithm for B-splines (link) results in the same as convolving block waves (e.g. functions defined as 1 for $\displaystyle x \in [0,1]$, 0 for $\displaystyle x \notin [0,1]$) ?

Both seem to result in the same thing (e.g. convolving block waves results in the same square wave as using the recursive algorithm), and both converge to a Gaussian distribution.

Surely, there must be some kind of difference between them, no?