These two scatter plots illustrate the difference, the first is for the points $(C,S)=(\cos\theta_1+\cos\theta_2,\sin\theta_1+\sin\theta_2)$, the second for the points $(1+\cos U,\sin U)=(1+\cos\theta_3,\sin\theta_3)$, where all angles $\theta_i$ are uniformly distributed in $(0,2\pi)$.

The second distribution uniformly fills a unit circle with center at $(1,0)$. The first distribution fills a disc with radius 2, but not uniformly, the density diverges as $1/r$ at a distance $r$ from the origin.

To calculate the radial distribution $P(r)$, it is helpful to think of a random walk on the plane with unit step size and random orientation. The desired $P(r)$ is the distribution of the distance from the origin after two steps.

After the first step the random walker is at some arbitrary point on the unit circle. For the distribution $P(r)$ it does not matter where on the unit circle, we may place the point at $(1,0)$. Then the second step brings the random walker to the point $(1+\cos\phi,\sin\phi)$, with $\phi$ uniformly in $(0,2\pi)$. The distance from the origin is $r=\sqrt{2+2\cos\phi}$, and the probability distribution is
$$P(r)=\frac{2/\pi}{\sqrt{4-r^2}},\;\;0<r<2.$$
The corresponding density $\rho(r)=P(r)/2\pi r$ indeed diverges $\propto 1/r$ when $r\rightarrow 0$.

For the generalization to $(C_n,S_n)=(\sum_{i=1}^n\cos\theta_i,\sum_{i=1}^n\sin\theta_i)$ see this MO posting.