* Transmission Differences: While both are likely to be 4L60E transmissions (a common transmission for those years), there were revisions and updates to the 4L60E throughout those years. Internal components, valve bodies, and even the bellhousing (the part that connects to the engine) may have subtle differences. A direct bolt-in is not guaranteed.
* Computer Compatibility: The transmission control module (TCM) in the 2003 Silverado is calibrated for the specific transmission installed that year. Using a 1999 transmission might cause incompatibility issues with the TCM's programming, leading to shifting problems, poor performance, and potential damage. You'd likely need to have the TCM reprogrammed, which is not a simple task.
* Other potential issues: Even if the bellhousing bolts up, there could be differences in the torque converter, input shaft, or output shaft that would prevent proper installation or operation.
In short: While some parts *might* be interchangeable, attempting this swap without a deep understanding of automatic transmissions and their compatibility across model years is risky. It's far more likely to result in problems and extra costs than a successful, trouble-free installation. Finding a used transmission from a 2003 Silverado is a much safer and simpler option.