How to using a 3.5 mm stereo plug mic with a EM11G LM 386N-1 amplifier
The first mic appears to have an integrated preamp, which presumably amplifies the microphone level signal (very low amplitude) quite a bit, giving you something closer to a line signal, which then drives the 386.

As for the second mic, note that one of the reviews seems to indicate that the microphone is an active microphone (and therefore requires a power source) but does not come with a power source. It is expected to be connected to a device that provides power. It might need a few volts, it might need full phantom power, or it might need some bizarre arbitrary voltage supply.

Alternatively, you might have shorted it or otherwise miswired it, because there is nothing that says a microphone will have the same connection scheme as a pair of headphones. Unfortunately, there is no standard for wiring a microphone to a 3.5mm jack (e.g. stereo, tip/ring bridged, DC supply on ring, balanced, and so on).

Edit: even if the second microphone doesn't require power, you need a mic preamp stage. The output of a typical passive microphone is ~50dB below line level. Your 386 circuit is entirely insufficient for amplifying that small of a signal; you will need to build a mix preamp. 

In short, there are many things that could be going wrong...