There were some suspicions that Nintendo would be recycling hardware from the NES Classic for the NES Classic and it turns out they were right. It's not too surprising that a console built to run game classic emulations didn't need to see an update for the following iteration, but it's interesting to see Nintendo using the exact same parts in the two.
Digital Foundry tore down the SNES Classic and compared it to the innards of the NES Classic and discovered that the two consoles came packed with the same exact hardware. It's not even that they are using similar hardware – it's the exact same.
The internal motherboard for the NES Classic had to have its corners cut to fit within the NES shell and the SNES Classic, which has a lot more room, has the same motherboard, cut in the same places making it a somewhat odd fit.
Below you can see the SNES Classic motherboard on the left and the NES Classic on the right. Note the corners.
As for the hardware, the two feature the same Allwinner R16 SoC (system on chip), with four ARM Cortex A7s and an ARM Mali 400 MP2 GPU. The single memory chip comes from Hynix and is a 256MB DDR3 module. On top of that, there's 512MB of NAND storage.
With so many similarities, you can expect the SNES Classic to get hacked to feature new games fairly quickly, since the NES Classic has already been cracked.