Looking at the specs of the ECC chip, it has support for asymmetric (public/private) keys using (ECDSA), and storage for certificates. The ECC chip is a superset of the ATSHA204 chip. So it has SHA-256 and HMAC/SHA-256. (message integrity/authentication)
The ATAES132 has support for AES (confidentiality)( and AES-CCM (confidentiality and authenticity).
Both chips have a hardware random number generator.
In short, the ECC chip is great for signing (authentication), and the AES is great for encryption (privacy) but it does have the AES-CCM which is used to authenticate/encrypt.
it seems like the two chips are complimentary in purpose. You cannot encrypt with the ECC chip, and you cannot use PKI with the AES chip.
If you want privacy, you need the AES chip. If you want strong per-device authentication, then you need the ECC chip.
Both can ensure a stranger cannot send bogus messages to your device.
If you use the AES chip, you can provide authentication with pre-distributed keys. The question is - do you need per-device authentication or not. If so, you may have a scalability issue. You need (N*(N-1))/2 keys for N devices i.e. 4 devices->6 keys, 5 devices -> 10 keys.
But you might not need per-device authentication, if you just want to have a single shared key with all devices, then you just need one or perhaps two keys.
And you also need to get these keys installed somehow. A single shared key can be part of the firmware, and frankly this is the simplist implementaiton.
I also think having a AES accelerator would provide longer battery life. Frankly, the AES chip would be MUCH easier to integrate into flutter.