The differences are:
1) Different display/touchscreen. The original display was no longer manufactured, so we replaced it with a panel from a different vendor. The new panel is slightly brighter and some people seem to think the colors are bit more vivid, but that's a pretty subjective thing, and to me it's really only visible when they're set side-by-side with the same content. This was the primary reason for the new version - we had to alter the motherboard for the new display anyway, so we addressed a couple of other issues while we were at it.
2) Change to the crypto processor to power up whenever power is applied to the device.
3) The backlight for the display is controllable, so you can change the brightness of the display beyond the two fixed levels. This manifests in the Control Panel as a different UI when you select Settings->Brightness. In 3.7 hardware, there are two radio buttons to set the display to full bright and dim, while in 3.8, you are presented with two sliders to set the normal brightness and the "night mode" brightness. I believe this is the only place where the Control Panel differentiates between the two models.
4) There is a different affordance for the attachment of charms. The 3.7 devices use a rivet, while the 3.8 devices have a little plastic attachment. Both 3.7 and 3.8 devices also have a little magnet on the side opposite the charms. It's amazing how few people have noticed that - it's a little hardware "easter egg" we threw in.
5) We have an additional antenna wire on the wifi adapter, which may increase the range of the device for some people. The geometry of your environment is still the biggest factor, however.
The USB, audio, processor, memory, etc are the same.
The devices run exactly the same firmware, widgets, Control Panel, etc - the drivers compensate for the minor differences in the brightness subsystem and default touchscreen calibration.
I'll remind bunnie to post the 3.8 schematics.