TL;DR Headphones will make a much larger difference than amplifiers with your current setup.
In most modern audio setups, you have a source, DAC (digital to analog convertor), an amp, and finally the driver (headphone or speakers). There's no need for a DAC if you're already using an analog source such as a turntable.
For most, the source is a computer or a media player. Both these have built in DAC's and amplifiers, but typically of mediocre quality. More on that later. Anyways, the source reads the music file, and outputs a bunch of ones and zeroes. The DAC takes the 1's and 0's and changes it to an analog signal that can be played. This signal is boosted when it passes through the amplifier, before the driver finally turns it into soundwaves.
In the case of the amplifier, it merely amplifies the signal (big surprise). This is important because some drivers require a lot of power to be driven properly. In your case though, both headphones are relatively efficient and do not really need the power of a dedicated amplifier. That being said, amplifiers also vary in how accurately they boost the signal. That is where most internal amps fall short.
However, to take advantage of this, you would need to bypass the internal amp of your source. This can be done via an external USB DAC or with a LOD (Line-Out Dock) for iPods. Otherwise, you are just feeding a dirty signal to be amplified to a more powerful dirty signal.
In addition, the amps you have mentioned will not make a very large improvement over the amp in an internal source; they are better for driving demanding headphones. They do have an EQ, but so do most music players.
Headphones will be a much more effective upgrade for you at the moment. The Hesh are decent, but there are much better that can be had for around the price.