A double-pole breaker with a "60" on each half will trip at 60 amps. The reason it's two connected breakers is to give the circuit 240V potential, by using two opposing AC phases. These do not "add up" to more current -- you're using them to double the voltage.
So if the heater requires 240V power at 120amps, you will need double-pole breakers that can provide a total of 120amps, e.g. 4 30A double-pole breakers as Tester101 suggested.
An electrician will be able to tell you if your service panel can accommodate this extra load -- it's a lot, but your 200A service might be adequate, depending on the normal load elsewhere in your home.
Wiring from garage panel to welder
NEC 630.11(A) may be of interest.
National Electric Code 2008
ARTICLE 630 Electric Welders
II. Arc Welders
630.11 Ampacity of Supply Conductors.
(A) Individual Welders. The ampacity of the supply conductors shall be not less than the I1eff value on the rating plate.
Alternatively, if the I1eff is not given, the ampacity of the supply
conductors shall not be less than the current value determined by
multiplying the rated primary current in amperes given on the welder
rating plate by the factor shown in Table 630.11(A) based on the duty
cycle of the welder.
Also take a look at 630.12 Overcurrent Protection.
The Owners manual (PDF) seems to contain the answer.
Wiring from service panel to sub panel
If we look at 110.14(C) in 2008 NEC, it says:
National Electrical Code 2008
ARTICLE 110 Requirements for Electrical Installations
I. General
110.14 Electrical Connections.
(C) Temperature Limitations. The temperature rating associated with the ampacity of a conductor shall be selected and coordinated so
as not to exceed the lowest temperature rating of any connected
termination, conductor, or device. Conductors with temperature ratings
higher than specified for terminations shall be permitted to be used
for ampacity adjustment, correction, or both.
Which means we'll have to check the breakers temperature rating, and the subpanel terminals temperature rating. Then we'll have to look at table 310.16 (Table 310.15(B)(16) NEC 2011), so we can determine the wire size we should use.
In this example image you can see if we're using copper, we'll use 60°C. Once we look at the table, we'll see that for 50A @ 60°C we need 6 AWG copper wire. (50A @ 75°C = 8 AWG).
110.14(C)(1)(a) basically says that if the temperature is not listed, use 60°C for circuits less than 100A. 110.14(C)(1)(b) says if the temperature is not listed, use 75°C for circuits over 100A. However, in the real world 75°C is often used for both.
Best Answer
They are the same thing and you don't have a choice. Your breaker is going to be a 2-pole breaker rated 15A. When you look closely, it will very much resemble two 15A breakers bolted together. That's not far from the truth.
The breaker will mount in the service panel in two spaces, which could otherwise accommodate two 120V breakers. What is happening is there are two 120V buses in the panel, and the 2-pole breaker is designed to tap both of them. They are phased opposite (when one is -120V the other is +120V), so if you tap both, you get 240V.