SD, SDHC, and SDXC are all electrically the same (though newer cards can also optionally support higher-speed interface variants with different signalling voltages), it's just that the original SD protocol didn't have enough bits in the protocol to support higher capacities. SDHC and SDXC are basically the same except that one uses FAT32 and the other uses exFAT as its standard format. You can often use cards above the official 32GB limit in SDHC devices if you can format them as FAT32. A lot of older SD card slots can also be made to support SDHC with a suitable firmware or driver upgrade because the protocols are really similar, but with some fields just being interpreted slightly differently.
