Let $C_1$ be a hyperelliptic curve of genus $g \geq 3$ (example: $y^7 = x^2 + 1$ for $g = 3$), and $C_2$ be a non-hyperelliptic curve of the same genus $g$ (for example, the Klein quartic with $g = 3$ again: I'll use it in the form $y^7 = x^2(x-1)$).

Then let $K_1$, $K_2$ be the canonical divisors of $C_1$, $C_2$ respectively (In the example above: $K_1 = 4[\infty_1]$ and $K_2 = 4[\infty_2]$, where $\infty_1, \infty_2$ are the "points at infinity" on the curves above: note that the projective curves corresponding to the equations above both have singularities at infinity, but in both cases these singularities can be resolved to give a single point at infinity on each curve.) Then $K_1$ is not very ample but $K_2$ is: this is a standard application of Riemann-Roch (see also Hartshorne, Chapter IV, Proposition 5.2.) The reason that $K_1$ is not very ample is that the map $C_1 \rightarrow P^2$ associated to $K_1$ is a double cover of a rational curve (a conic, in fact) in $P^2$, rather than an embedding.