When you beam up someone, are you beaming up the matter or the information about the matter (i.e. - The bits or the atoms?)
Why choose? By beaming slightly "de-matterized", slightly "informationalized" goo, you get the best of both worlds: the structure of the matter carries some of the information needed, and its halfway nature reduces the costs of disassembling and reassembling.
This is what the dialogue (and visual evidence!) suggests, too: matter is turned into "phased" form, which is invisible and goes through walls and moves at lightspeed and whatnot, but maintains internal coherence and carries information.
If you were beaming up the atoms, then consider you'd have to have enough energy to break down every atom into pure energy (the equivalent of many, many hydrogen bombs.)
But you'd get it back at reassembly. So you'd just have to optimize. That's how you build an industrial plant today, too: some parts of a process require heating, others require cooling, and you carefully balance the energy budget before designing the actual pipes and radiators so that you make maximal use of "free" heat. With transporters, disassembly and reassembly might take place simultaneously, with a net energy consumption approaching zero.
In slightly delayed processes, unless you have access to time travel technology (which Starfleet has!) you just have to "realize" those immense energies for a while (which Starfleet regularly does, with the warp drive and all!).
On the other hand, storing the info on 10 to the 28th power atoms in the human body is no cakewalk either!
It's not a matter of storage but of transfer. If it's not instantaneous, then we're dealing with the transfer of finite amounts of information per unit of time, possibly across multiple channels. Might be quite manageable even with today's technologies, really.
Although I'm rather partial towards the "best of both worlds" solution. Even today, by far the fastest way to get lots of information from New York to Paris is to only digitize the text and the numbers, put that data in a memory stick, attach the pictures in physical form, and send the envelope to Paris via a courier service...
Timo Saloniemi