Have problems running a 8 ohm, 89 db speaker

I bought a minisystem from overseas, those that have a 6 ohm amp, that is supposed to be able to drive things that are more efficient than the speakers provided 81db efficiency, 6 ohm two way vinyl clad speakers. Well, I have this minisystem as a source system, has 3 CD charger and a MD recorder, with its power source being a 220volt to 110 volt transformer. I tried to run the EPOS ES 11 from its amp section and the only thing I get is very bad quality sound. I remember this system really pounding the 6 ohm, 81db it had come with, and now, I am wondering. Does this driving issue have to do with the fact that the amp section of this minisystem, which is rated 25 watts at 6 ohm, can't produce enough watts to drive something about that impedance? I always notice that amps produce more watts as they go down while their wattage goes down as impedence goes up. What do you think is the problem with this set up?
Perhaps the electronics in the minisystem were designed with equalization built in to maximize the sound from the speakers that came with it and the speakers you are trying to use don't need it. The power should not be an issue. If anything, the new speakers will be an easier load.
"25 watts at 6 ohms" means nothing. What are the bandwidth/distortion figures? Or aren't they even quoted in the specs? In general, minisystem specs are sketchy and highly imaginative. I'm surprised the speakers even had a sensitivity spec. At any rate, you're trying to do something a minisystem really isn't designed to do, which is drive a set of speakers other than the ones included. I recommend you give that up.

Your observations about watts and impedance are normal. Almost all amps will deliver more power into lower resistance.
Years ago, one of my sons took an unused KLH5 speaker that I had, and hooked it up to his little table-top FM radio. Now the KLH5 was a relatively low efficiency quality item, and the little radio probably had less than 3 watt audio output. I told my son it was ridiculous, and he wouldn't hear anything from the speaker. Was I ever wrong. Amazingly, it sounded almost good, if not loud. It may not make sense to use a quality speaker with electronic junk, but I ought to work, and make the junk sound a bit better.