The other answer is on the right track, but doesn't emphasize the real difference (IMO).
Let's say it takes 100 hp (or kW if you prefer) to compress the amount of air you need for a given purpose. The jet engine already has a very efficient compressor, it would be hard to do better with a separate compressor. Let's assume it takes the same power to do the job either way.
In a bleedless system, we have a generator connected to the engine that pulls off 100 hp of shaft work and converts it to 100 hp of electricity. We use that electricity to run a motor to compress the air. These devices each have an efficiency (power loss), but that isn't the important part. Fundamentally, that power comes from the turbine, it was extracted from the engine cycle.
In a bleed system, we pull a certain amount of compressed air out of the compressor ($\dot{m}_\mathrm{\,bleed}$). The compressor runs off of the shaft connected to the turbine, so it still takes 100hp of power from the turbine to compress the air. Roughly speaking, the power extracted from the engine cycle is the same.
However, in a bleed system, you're also stealing air from the cycle. Without bleed, that $\dot{m}_\mathrm{\,bleed}$ would continue to the combustor, would be mixed with fuel, burned, and then to the turbine where energy would be extracted (to help turn the compressor and other accessories) and then expanded out the nozzle producing thrust. I.e. the point of the engine cycle.
The air at the end of the compressor not only took power to reach that state, but it still has an intrinsic value to the engine. The more air flow through the engine, the more ability it has to produce shaft power and thrust. When you steal some of that air flow, you're penalizing the cycle.
In effect, bleed air puts two penalties on the engine (power extracted and mass flow stolen from the cycle). Whereas a bleedless system only puts one penalty on the engine (power extracted).
Most bleed loads do not operate at all times. You need a lot of bleed air during engine startup, but less at cruise. Some is a constant load (the bleed air used for turbine cooling) and some turns on and off -- air conditioning, engine start, etc.
An engine designed for a wide range of bleed across a wide range of operating conditions (very little to a great deal of bleed from flight idle to cruise and takeoff power) will have a lot of accommodations in the engine control -- and will fundamentally be a bigger, heavier, less efficient engine. The bleedless alternative is an engine with a wide range of power takeoff at the wide range of operating conditions (few hp to many hp at all those flight conditions). This is easier, simpler, and more efficient to achieve.
Bleed air for turbine cooling is a relatively constant load (easier to design for) and also keeps the engine a simple, self-contained unit. It is safer to cool an engine with bleed air than to introduce a separate electric compressor to do the job. However, cabin air isn't the same level of criticality and it is easier to add redundancy outside of the engine to make sure everyone has air to breathe.
The 787's 'more electric architecture' tried to move many systems to electric. This was accommodated by a large starter-generator attached to the engine. Instead of using bleed air (first from the APU, but then from the #1 engine) to start the #2 engine, it uses electricity (first from the APU, but then from the #1 engine) to start #2. This allowed the removal of a huge amount of high pressure pneumatic lines from the tail of the aircraft, out the wings, and to each engine -- and from one engine to the other. Those ducts were replaced with copper wires.
Engine start is the largest bleed load -- once that was removed (and replaced with sufficient capacity in a starter/generator (one electric machine (read mass) that does two jobs), then it makes sense to try to do other jobs using electricity instead of bleed.
So, while removing bleed air is a good thing for the engine -- this is also a systems wide tradeoff that can't just be considered in the context of cabin air, but also all the other jobs that compressed air (and electricity) do on the airplane.