Both have been interpreted differently in different contexts and there is no standardized definition across the globe. Here we explain what it is and the differences between both.
Here, your mobile acts as an access channel for your existing banking relationship. This implies that you have an existing relationship (or in some cases can apply for an account) while using the bank's mobile banking system. You can carry out basic functions like funds transfer, pay bills, check mini statements, view balance, apply for loans, apply for checkbooks, stop checks etc. Note that all these transactions are based on your existing bank accounts. The service would vary depending on what kind of account you have - savings, current, loan, credit etc.
Mobile money is mobile based accounts offered primarily to the underbanked and unbanked. It is a service offered by the mobile network operators to their subscribers. Thus they typically don't have any banks as a part of the ecosystem (though you do find banks getting into this business in emerging economies of late). The type of transactions you would see on a mobile money system will be prepaid airtime recharge, domestic and international remittances, merchant payments.
Mobile payments are any kind of payments initiated through a mobile phone. The source of funds could be anything - cards, bank account, mobile money (stored value account), prepaid account etc. It could be a part of mobile money, mobile banking, mobile wallets or any such service.