How can relays be rated at 10A for both 240VAC and 120VAC?
There are a large number of design elements that affect the current and voltage ratings, technically, and then there are the approvals they choose to get. Many relays have multiple ratings, depending on the safety standard to which they are tested. 

Generally "watts" is not the limiting factor, it's amperes for heating, and the voltage for contact opening. The wear on the contacts can also be limiting and that's affected by factors such as the inrush current of the load and any reactance (eg. inductance) in the load. 

Generally small relays are rated for some number (100,000 is typical) life at full load, and you can derate that (eg. operate a 20A relay at 5A) to get a better lifetime. 

It's not that common to find a drop-in relay that will give you substantially more life or current rating with the same load and contact metallurgy-- relays are fairly well optimized. You'll usually find the higher rated relay will draw substantially more coil current (perhaps that your existing circuit cannot safely handle) or will be substantially larger or both. 

Your best bet may be to operate a larger relay or contactor with your small relay.