If the Earth's atmosphere is rotating at the same speed as Earth, then the atmosphere must be rotating much faster at the equator than at the poles. If you spin a ball covered in oil, it will form rings. Also Jupiter has rings. So why doesn't the Earth have rings of weather too?
The short answer is -- there are bands! They behave very similar to the bands on Jupiter, but are not as pronounced. And we don't have a really unappealing colored atmosphere to show us what the bands look like.
Here is an example of what they look like (source):
There are two bands along each side of the equator. Another set of bands starts 30 degrees north and south of the equator. And another band starts 30 degrees further north and south (at 60 degrees total).
You'll also note that these differences in wind in the same direction of rotation also causes wind to form in the north-south direction. All of this is what drives the major weather systems.
Consider the US. Weather systems will typically move from west to east. Atlantic hurricanes form in the tropical band off the coast of Africa. They form here because the wind is relatively calm and there is little north/south shearing. They then move westward in the tropical band while also moving north due to the Coriolis forces. As they move north, they begin to encounter the westerly winds that are characteristic of the mid-latitude cell. This will eventually turn them around so they move north-east along the US coastline until turning due-east and moving towards Europe (which in turn induces a southward drift due to Coriolis forces).
Here are what several of these hurricane paths look like (source):
These bands are not typically readily apparent. Mostly this is because our atmosphere is transparent so we have no way to "visualize" the bands. It is possible to sometimes capture bands however. A band of rainfall in the intertropical convergence zone around the equator is captured in this GOES satellite image (source):
Also, these bands are climatological features and not meteorological features. This means their structure is not always apparent instantaneously but appear in a time-averaged view of the atmosphere. It turns out that NOAA released a time-lapsed video of 10 years worth of GOES-12 images and the bands become pretty apparent!
@DavidHammen found another great video looking at the infrared signature caused by water vapor in the air by the GOES-13 satellite shows the bands better than looking at the visible cloud cover.