Airlines periodically devalue their points programs
Not every destination included in award travel
Have to book far in advance for popular routes or holidays
Still have to pay applicable taxes on award fares. This can range from $5 to $300 depending on whether you are booking with the airline directly or via alliance airlines in the network (like British Airways which can be a bit costly on taxes).
Bank Cards (Points)
These are travel cards from the financial institutions themselves like Chase Sapphire, Barclaycard Arrival, Capital One Venture, American Express Premier Awards, etc. These, on the other hand, give you:
Flexibility in award travel (any airline, any date)
Fair market price (you can book the cheapest fare)
Discounts on redemption (i.e. Chase Sapphire Preferred gives you a 23% discount on market fare, Chase Sapphire Reserve gives you 33% discount)
Ability to earn miles with for the flight since the fare is treated as "cash" by the airline
Protection against mileage devaluations with any particular airline
Many offer flexibility to transfer points to hotel points or airline miles
Often better customer service
Miles only accrue with spending (not flying)
However, they do offer an online shopping portal just like the airlines which is often more generous in points per dollar
No "first class" treatment by the airline such as priority boarding or a free checked bag unless you already have status.
Need excellent credit
Often can't "churn" these cards as easily
Having carried and redeemed award travel on both types of cards, my strong recommendation is to use a bank points card and then slowly churn your airline and hotel cards to take advantage of the signing bonuses. At any point, you should have two cards in your wallet: your evergreen bank points card and your latest airline credit card.
The goal is to avoid paying an annual fee for the airline cards, while ultimately, the benefits of the bank cards (flexibility, fair market price, points transfer, discounts, customer service, etc.) are worth making it your primary card. These cards are worth the annual fee and are hard to churn since their credit standards are often higher. I consolidate my points with Chase Ultimate Rewards. Here's why.
Bonus tip for the airline cards: Credit cards can have many authorized users, so to get some of the frills such as a checked bag and priority boarding, consider asking a spouse, family member or close friend to add you as an authorized user to their airline card. This accomplishes the best of both worlds and on some elite cards can even get you into lounges carrying these cards.