While it is true that, because of the (so to speak) mechanics of online commerce, the two can often be used interchangeably, there is a difference in what the terms represent. Ownership officially changes with a purchase; but not necessarily with an order.
An order is the result of a performative verb to order. If I order you to shut the door (assuming all the felicity conditions for ordering are true), then you have an order to shut the door -- but I do not yet have a shut door -- once again assuming the context is felicitous, I can expect to have it soon, but I don't have it yet.
In a commercial context, the customer orders (in the particular commercial sense) the merchant to deliver a product, in return for payment. The customer pledges the payment, and the merchant pledges the product, but this is still at the pledging stage; nothing needs to change hands in an order, except by pre-arranged convention. Both sides can expect the pledges to be redeemed in the future; but not yet.
Purchase, on the other hand -- a higher-register term for buy -- refers to the entire Commercial Transaction Frame. If I purchase something, then I have already paid for it, reserved it, and own it. It may or may not be in my physical possession -- it might be in for repair, or not delivered yet, or loaned to a museum -- but it's mine and I possess it already.