There's no flaw in your argument. A computer heats the room just as effectively as an electric heater of the same power and you could use the computer to do something useful (Bitcoin mining?) while it's heating your room.
There are some practical considerations, though I think these have been sufficiently discussed in the comments. Computers would make for noisy and bulky heaters, though anyone who has had to refrigerate a server room will tell you they can be very effective heaters.
If you want to be really, really, pedantic the computer may not be quite as effective as a heater because an infinitesimally small amount of it's power may go into the data on its disk drive. See Is a hard drive heavier when it is full? for a discussion of this, but note that I mention this for its amusement value rather than because I seriously believe it's an issue.
Generally, bodies can have same internal energies, but have different temperatures, and vice versa, have the same temperature but different internal energies.
Consider for example ideal gasses, where the internal energy is given as a function of temperature and heat capacity: $U={C}_{V}T$. If we have a monoatomic gas consisting of N particles, its heat capacity is ${C}_{V}=\frac{3}{2}N{k}_{B}$, whereas for a gas consisting of diatomic molecules we have ${C}_{V}=\frac{5}{2}N{k}_{B}$ at room temperature (if you want to know where I got these explicit formulas, google the equipartition theorem). If you try to play a little with inserting different temperatures or plot those relations, you get something like this:
This graph is plotted for $N={N}_{A}$, namely for 1 mole of ideal gas.
Notice that you may have gasses with equal energy, but different temperature, and having the same temperature doesn't mean the internal energy of two systems is the same.
At temperatures below 100 K, the equipartition theorem doesn't hold and the heat capacities of both gasses slowly saturate to 0 at 0K (a consequence of the third law of thermodynamics). At higher temperatures (around 1000K), the heat capacity of the diatomic gas increases further because of vibrational degrees of freedom. Around room temperature the relation is linear, as shown.
As for your second question, assuming the two systems are in contact and able to transfer energy between them, they will continue to transfer energy between them until their temperatures are the same (that is, until they reach thermal equillibrium).
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This is the crucial assumption. If both have real power 2000 W, this means 2000 W of heat introduced into the wires and then (by assumption of stationarity of temperature of the heater) this heat leaves the wires and goes into the heater environment.
But equal heat introduced per unit time does not mean equal effect on average temperature or equal efficiency of operation. Bigger heater with greater exchange surface will operate on lower temperatures and will heat up the air more evenly quicker (because of energy losses from the room). This can be more efficient if we want to achieve same temperature everywhere as fast as possible, or less efficient, if we want to heat up only our frozen legs.
Also in practice, even if two heaters have 2000 W written on them, this does not mean that actual consumption is 2000 W all the time. For example, as the heat producing wires get hotter, their ohmic resistance increases, so the effective value of current should decrease and real power should decrease as well. The only way to be sure of the consumption of the heater is to measure it with a wattmeter.