Quite honestly, the plant doesn't know the difference between the two. It's only interested in the ions it can absorb through the roots and leaves. But choosing the right fertilizer and applying it correctly has a significant impact on the hardiness and vigor of your plant. For the purpose of discussion, synthetic fertilizers are those that are nitrate-based. Often, they contain ammonium nitrate.
Nitrates are the form of nitrogen used by the plant.
Conversely, organic fertilizers are nitrite-based and must be broken down by microbial activity. Nitrate-based fertilizers can cause serious damage to roses if applied incorrectly or too early in the season. These fertilizers often cause a spurt of tender, young growth - growth that has not had sufficient time to harden before winter's freezing weather is complete. Die-back is often the result. Likewise, applying these fertilizers creates an over-concentration of nitrogen in the growth tips.