I had this dilemma at one point when I had CBA Platinum cards (free when I used to work there) and Amex Platinum Edge.
I closed by CBA Platinum cards when they devalued their Award points for Frequent flyer program transfer and charged annual fee.
I hate paying annual fee out of principle. But weigh the benefits when it comes to paying them.
I've read a lot of points management experts/analyst on how to spend your points and most of them value Amex issued Amex a lot...
I love to travel a lot for leisure especially international doing the backpacking experience (4 weeks annual leave bounded).
When I refer to Amex issued Amex, I refer to Membership Rewards Ascent/Ascent Premium which comes with their charge cards, gold or platinum credit cards (exclude David Jones). This excludes the co-branded cards as well e.g. Qantas and Velocity.
Advantages on Amex issued Amex:
1. Membership Reward (MR) points are worth a lot more than bank issued Amex especially when you do conversion to Frequent Flyer programs/hotel programs. Points expert value them at least 1c per 1 MR.
2. Flexibility on transferring the points to frequent flyer programs (Emirates, Etihad, Singapore Airlines, Malaysia Airlines, Cathay Pacific, Virgin Atlantic, Virgin Australia, Air NZ, Thai Airways) /hotel programs (HHonors, Starwood Preferred Guest, Jumeirah) of your choice. I use the MR points to convert to either Velocity or MH Enrich depending on promo transfer which comes up at least once a year for them about 15% bonus miles extra. Conversions are usually 1 MR to 1 FF. Hotels vary a little bit such as Starwood Preferred Guest (SPG is a really good hotel program especially the conversion of their hotel points to other FF programs such as AAdvantage, Dividend Miles etc.).
3. Amex customer service is far superior to banks here. They go the extra mile to help you sort out your issues especially card replacement, stuck in a foreign country, doctor speaks your language etc. Unfortunately, there are unsatisfied experiences but I have not received so far.
4. Amex run promos/competitions to boost MR points is really fun to participate.
5. Free Virgin Australia domestic flights on the platinum credit cards (Edge, Reserve) are worth the annual fee especially the flight price outweighs the annual fee charged.
6. Amex card replacement is super committed. They come visit you personally if they have to in any part of the world to give you the replacement.
7. The tiered MR points earn structure works depending on your spending habits.
Disadvantages on Amex issued Amex:
1. Travel insurance cover (gold/platinum cards) is not unlimited medical cover compared to bank issue cards. When it comes to medical needs, it's better not to put a limit on it. Go buy a separate travel insurance e.g. Travel Insurance Direct or anyone else.
2. 0.5 MR points earn on government charges, insurance (except Amex issued insurance), utilities, telecommunications.
3. Australia charges one of the highest annual fee in comparison to Amex US (compare Platinum Charge)
4. To get annual fee waiver is a little hard even with desire to cancel the card. They rather you cancel the card than give you a fee waiver. If the lucky fairy visits, good on you.
5. They have discontinued using MR points to redeem for goods e.g. electronic things. That's ok, as it is a poor form of redemption to get good value out of them.
6. They still don't have a no foreign currency conversion fee card. I have personally lobbied to Amex to introduce such a feature especially for premium card holders...
7. Limit on maximum points earn for Platinum Charge and Reserve cards (tiered structure), then reduce to 1 MR to $1 on purchases.
8. Qantas points conversion requires you to register your QF card number which gets sweeped monthly. You lose the MR benefits e.g. promo points earn.
Advantages on Bank issued Amex:
1. Travel insurance varies for different banks but there are unlimited cover.
2. Points are earned at their maximum usually about 3 points per dollar but you're at the mercy of the banks conversion rate to frequent flyer programs which can change over time if they decide to devalue their points. 3 points can give you about 1.5 miles (e.g. Westpac Altitude points)
3. You can use the points to redeem for goods e.g. electronics (still it's a poor value redemption). Consider gift cards option instead.
Disadvantages on Bank issued Amex:
1. Frequent flyer programs points transferred are limited to a few around.
2. Hotel programs are non-existent for points transfer as far as I have seen (correct me if I'm wrong)
3. They come with Visa/Mastercard which has poor points earn. There are standalone Visa/Mastercard which has better points earn e.g. Citibank Visa Signature, Bankwest QF Platinum.
4. Limits apply on number of points earn per year e.g. CBA cards, Westpac cards
5. No ongoing promo/hardly ever for extra points earn for existing customers.
6. Annual fees on cards are also quite high for premium cards.
7. Banks customer service is not global enough unlike Amex (searching for doctor that don't speak your language kind of commitment)
8. Card replacement probably wait till you get back to Australia to get it.
Here are my 2 cents. There may be more to advantages/disadvantages. Hence, I value my Amex issued Amex.