Why some differential equation can be solved while similar difference equations cannot? Take an equation
$$w'+w-w^2-1=0$$
Its solution is 
$$w(x)=\frac{\sqrt{3}}{2} \tan \left( \frac{\sqrt{3}}2 C+\frac{\sqrt{3}}2 x\right)+\frac12$$
I wonder why a similar difference equation 
$$\Delta w+w-w^2-1=0$$
cannot be solved?
 A: UPDATE :
Let's start by showing a solution of the difference equation : 
$$\Delta w+w-w^2-1=0$$ 
at least if this means $\ (w_{n+1}-w_n)+w_n=w_n^2+1$ because :
$$w_{n+1}=w_n^2+1$$
admits the solution (for the specific case $w_0=1$) :
$$w_n=\lfloor c^{2^n}\rfloor,\\\text{with}\quad c=\exp\left|\sum_{j=0}^\infty 2^{-j-1}\ln(1+w_j^{-2})\right|,\\c\approx 1.5028368010497564997529364237321694087388717439635793$$
This solution and others are specific cases of the quadratic map.
I'll add that other integer initial values of $w_0$ may be handled with this method (for $w_0=2$ for example we only need to replace $c$ by $c^2$).
Some may object that there is some kind of cheating here since $c$ is 'built' with the first terms but that's life... 
The original 1973 Aho & Sloane's paper "Some doubly exponential sequences" is available here (Wayback Machine)

I will now try to consider the real question with a little more care...
Let's first note that the differential as well as the difference problem are both nonlinear...
Non linearity means not only that general solutions can't be obtained just by superposition of a few solutions but often that closed form solutions will exist only for specific initial conditions (if they exist at all...).
I will not develop further the very interesting subject of dynamical systems and chaos that may appear for some systems (or parameters) but observe that $w_{n+1}=w_n^2+c$ with $c$ a complex number generates a fractal : the famous Mandelbrot set (see too the quadratic map link). As pointed by Gerry your differential equation was separable (and solved) but just adding $+i$ at the end adds some interesting effects...
At this point I must admit that all these digressions still didn't answer to your question since you noted the O.D.E.'s solution and asked why the difference equation could not be solved. In fact the difference equation can be solved : start with $w_0$ and use the iterations to get the next values. But it is clear that you wanted a closed form for the nth term. 
The difference equation is often used as an approximation of the differential system but the dynamic is only similar at a small scale : the $\tan$ goes to $\infty$ after a fixed time, the difference equation is squared at every iteration and will need an infinite time to go to $\infty$. 
Another nice illustration of these differences may be seen at Wolfram : the ODE gives circles while the difference equation gives spirals (the shorter the steps the nearer we will be to a circle but it will never be a circle...).
ADDITIONS :

Let's interpret the picture :


*

*in the differential case : you get very simple regular orbits in the phase plane ($x,\ x'$) (it is a one dimensional problem with the position at the bottom and the speed on the vertical side). Physically this may represent one dimensional oscillations with various amplitudes (in the middle is a fixed point with no oscillation at all). So we have the simplest system with energy conservation (oscillations will never stop nor change).

*in the difference case : you get orbits that look locally a little like the differential case but with a very different topology : orbits are not closed on themselves, oscillations will amplify without end as time passes, energy is not conserved but increases (decreases) instead (except for the central fixed point)! In fact this is a frequent problem with more elaborate numerical simulations (using difference methods or their more sophisticated counterparts like Runge-Kutta, predictor-corrector and so on...) : the model becomes hotter and hotter (simulated galaxies lose their stability and explode...). Going from continuous to discrete systems must always be handled with care! 


Note that sometimes the discrete system will be much simpler than the continuous one, in fact they should simply be considered as different (globally even if locally as similar as wished) so that even the existence of an analytic solution in one case will often be unrelated with any closed form of the other!
I hope all this helped more, 
