In terms of how it effects the meaning of the phrase they're used in, they're the same thing, but "clash" provides descriptive imagery. A clash is a collision between two objects; especially a noisy collision. For me, it brings to mind a pair of cymbals banging together. So you can see how where "conflict" might state an opposition of two sides, "clash" might also convey the tension felt between the two parties.

This is a very difficult question that is one of nuance, I don't know if my answer is technically correct but this is how I would use it in conversation. A conflict refers to something general or ongoing. For example, there is a lot of conflict in the Middle East, or the conflict in the Democratic Republic of the Congo has been going on for decades. Clash refers to something immediate or a definite starting point in time and usually refers to direct fighting or violence. For example, the West and the East clashed today as they could not settle the border dispute. Another example that uses both would be: Both sides clashed early Wednesday morning, experts expect the conflict will last for months.