What problems can arise if two Founders of an early stage startup share the same equity, but one of them is appointed as the CEO?

This is personal opinion, more than an answer.

You really needed to decide all this when you first set up the company. You needed to discuss with your partner how much money he was bringing to the table, and if not money the value of his contribution to the company.

In the end CEO, COO, CTO, any other alphabet soup you wish to create are just titles, titles are used for talking wth other companies so that they know what your job is in an organization.

As you grow, you will maybe get a larger board of directors, these will be share holders, and in order to get these you will need to each part with initial A shares in the company, what you need to worry about then is you and your partner always keeping a controlling interest.

But at any time, if you or your partner have a falling out, one or the other of you could get the other members of the board to side with you, giving that board member the voting power to remove or buy out the other, that is just how businesses work. I give Steve Jobs as an example, he founded a company, was CEO, gave up the position of CEO to another, and was eventually voted out of his own company by the board.

If this is what is worrying you, then you are wasting your energy, energy that could be used to make your company great. So stop doing that, and remember this... The past is the past, all you can do is learn from it, the future is uncertain, and you have no real control of it, so all you can do is work on right now, and that is where you should be focusing your energy.

I am sure that others say that one person or the other should have controlling interest and that, I think depends on how they set up their businesses. You started this with a partner, someone I assume you like and want to work with, so why(in my opinion) are you try to gain leverage over your friend?

My last opinion is that perhaps you should add one more person to your board, only because you do not have a deciding vote if things really did come to a 50/50 split. This person does not really need to have any A stock, it just needs to be in your bylaws to say that in the event of a 50/50 split vote by the board this person has the deciding vote. In the future, this person could be the CEO because in most large companies the CEO answers to the board, and is not the controller of the board. But all of this you need to sit down and discuss with all your controlling partners. In your case that is the four of you.