The escape layer of the atmosphere is several thousand kilometers up, which is many times as high as the shield at about 200 km.
So the atmosphere above the shield would continue to slowly escape into space, but the shield would stop gas from lower down from moving up to the upper atmosphere. Thus the atmosphere above the shield would diminish faster than it does now. However, the slow rate of atmospheric loss should mean that will not be a problem in the long range, and certainly not during the time scale of most stories.
Micrometeoroids burn up lower than the shield, so people would no longer see meteor trails in the night sky. Instead space dust would hit the shield and be shattered or even vaporized by the impact, or possibly merely sit on the shield. Larger space rocks and asteroids would also be stopped by the shield, perhaps exploding spectacularly.
Since dust particles are opaque, the higher the percentage of the shield's surface which was covered by dust particles, the lower the percentage of the sunlight hitting the shield that would reach the surface of the Earth. I expect that dust buildup on the shield would probably be far too slow to be an important factor in the story.
The ionosphere (/aɪˈɒnəˌsfɪər/) is the ionized part of the upper atmosphere of Earth, from about 48 km (30 mi) to 965 km (600 mi) above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an important role in atmospheric electricity and forms the inner edge of the magnetosphere. It has practical importance because, among other functions, it influences radio propagation to distant places on Earth. It also affects GPS signals that travel through this layer.
So about a fifth of the Ionosphere would be inside the shell and about 4 fifths would outside the shell. If being outside the shell caused major changes in the Ionosphere it could affect long distance radio on Earth. But I don't think changes in the ionosphere would happen fast enough to affect the story, unless you are planning to write a multi billion year future history like Stapledon's Last and First Men.
So far I have not been able to think of any changes to the atmosphere that the shell at 200 kilometers would make happen fast enough for the purposes of a story. But if an expert in atmospheric physics can think of a change which would happen fast enough to be important for a story, they should mention it in their answer.