Taks,
In general, engines from both manufacturers are available for most commercial aircraft. Each engine manufacturer (i.e. GE, RR) designs the engines to meet the stated engine specifications from the aircraft manufacturer. This is in terms of engine size/weight, attachment fittings, thrust, fuel consumption, reliability, etc. Each airline is free to select an engine meeting those requirements from either GE or RR. Which engine they select generally depends on things like price, engine thrust and specific fuel consumption, which can be traded off depending on things like the routes to be flown for the particular airline. All of that being said, each engine manufacturer has their own proprietary techniques used in the design to eke out every possible ounce of performance. The internal guts of each manufacturer's engines will be similar, but not identical. For instance, the number, and dimensions, of the engine blades and the number and position of the blade assemblies will usually be different. Each manufacturer will design towards the same specifications, so given the same basic technology, the engines will be similar (at least from an external viewpoint) and will have roughly the same performance.
Hope this helps, although I suspect I've just managed to through a little confusion into things!