For an overview of the TMGC+C, read this topic:
https://www.tamatalk.com/IB/index.php?showtopic=157709
I've had a Tamagotchi-iD for only two weeks now, so haven't had much time to document all its features. Briefly here are some of the differences.
General
It seems most of the characters you can raise are different. The growth patterns are certainly different.
On the TMGC+C you can plant a tree outside in the yard, and harvest its fruit. There are no such trees on the Tamagotchi-iD.
On the Tamagotchi-iD there is a mailbox outside in the yard. About a day after you've connected your Tamagotchi-iD to another Tamagotchi (TMGC+C, Hexagontchi, or Tamagotchi-iD) your Tamagotchi-iD may receive a letter from them.
Happy bar / Friendship gauge
On the TMGC+C there are certain things you must find to extend the length of the happy bar. These things are the same for all characters. On the Tamagotch-iD these things are different for each character.
Meals & snacks
On the TMGC+C once you've bought a food, it goes into your Meals & Snacks menu, and your TMGC+C character can consume it at home (in their kitchen) without using it up.
On the Tamagotchi-iD, there is only one meal and one snack that can be consumed at home. For a larger variety of meals & snacks, your Tamagotchi-iD character must visit a convenience store, restaurant, doughnut shop, or cafe. Food consumed at these locations, must be purchased every visit.
Door (going outside)
You don't need transit passes (train tickets) to go to the various places. Places on the Tamagotchi-iD that are similar to the TMGC+C: game center, playground, home redecoration center, matchmaker's, and TamaDepa. New places on the Tamagotchi-iD: photo studio, accessories shop, and friends plaza.
The Tamagotchi-iD has two games in the game center. They're different than on the TMGC+C.
Accessories
You can purchase accessories (caps, bows, headphones, etc.) from the accessories shop, and have your Tamagotchi-iD wear them all the time (except when sleeping.) The TMGC+C doesn't have this feature.
The BIG Difference
With the Tamagotchi-iD you can download new mini-games for the game center, meals & snacks, accessories to purchase in the accessories shop, items to purchase in TamaDepa, etc. from tama-id.com via your cell-phone's infrared port. (It appears that tama-id.com works only for Japanese cell phones.)