a headphone amp generally will have different output specs to power headphones which have diff specs compared to speakers. the miniwatt can be used if you have a headphone output on the amp. i suppose there are some headphones that are dedicated and cannot be driven by normal outs from a discman or something like that. for these you'd definitely need a headphone amp.
I am not great in the technical stuff. Just my understanding. Both headphone amps and the normal amps are basically amplifiers. But the amount amplification done by headphone amps is very small mostly less than a Watt. Now, there are headphone amps that are powerful or even like your Miniwatt that produces a couple of watts that can be used for either purposes. But then the connectors are different and you need certain adaptors or cables to do the job. Also, its not advisable to randomly pick one and use for the other. Some are know to drive both and you need to know such things.
Otherwise the basic idea of a headphone amp is simple. Essentially the source , a CD player or an mp3 player after the DAC stage will send an analogue signal which will be very weak. This has to be then amplified for any speakers or headphones to reproduce them. This amplification is done through an output stage for an headphone. So, what you see in a player - the headphone socket - that is actually the output of an amp. But however most of the headphones we get are of the impedance 32ohms and easy to drive. so, the head amp outputs are decided based on this. If however you are using higher impedance headphones like 300 or 600 ohms, then the same player would not have enough juice. So, you pass this output to a dedicated headphone amp that can provide that juice.