Personally, I found the E11 a bit pointless, and it was only after getting the BH that I realised what an amp can really do. The bass boost is all the E11 has to offer; the amp doesn't enhance clarity or soundstage, it just enables you to get more volume. The BH, on the other hand, with its two outputs, lets you choose between a warmer and a clearer sound, and the switches let you modify bass and soundstage. The battery lasts forever, and you can use it while charging, which isn't the case with the E11. I highly recommend it over the E11.
It's also way more expensive for minimal upgrade. Sure I could spend a ****load of money for a Schitt amp, or some other amp with a boatload of features and buttons, but for the money and the sonic ability and features, the E11 IS A BETTER DEAL.