The iPhone will drive the headphone fine and make a very nice sound, but the amp will make a big improvement in harmonic extension and soundstage. Use the amp whenever possible. Both Headfonia and I concur that the E07k is a great amp, and probably the best thing you can get for up to twice the price. I’m familiar only with the E07k, the E17, and the E12. I think the E07k beats the E17 (and both are also USB DACs), while the E12 is just an amp, and has a darker sound but with more power for inefficient headphones. The Philips is not inefficient.
Wireless and truly wireless: These connect to your devices using Bluetooth, so you’re never physically tethered to your smartphone, tablet, or computer. Wireless headphones don’t use a wire to connect to an audio source, but they do use a wire to connect the two earpieces together. In contrast, “truly” wireless headphones come as two separate earpieces that don’t need wires to connect to anything. Wireless headphones are incredibly affordable; truly wireless headphones can cost anywhere from $100 to $400.
The Bose Frames are non-polarized sunglasses with special speakers built into each arm. The idea is that they’re designed to look like regular sunglasses, but also act as Bluetooth headphones. Since there’s no earbud that actually goes into your ears, the speakers have been engineered to shoot audio down into your ears; the neat thing is that the Bose Frames do a very good job at masking your audio so that the people around you can’t really hear what you’re listening to. They are available in two different frame styles, round (Rondo) or square (Aldo).
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Released in 2015, the SoundLink Around-Ear II are Bose’s wireless over-ear headphones that don’t have active noise cancellation. They’re lighter and slightly more travel-friendly than a lot of the company’s other offerings, and they could sound quality at the forefront. At the time, the big selling point for the SoundLink Around-Ear II was their sound quality — Bose claimed that they sounded as good as wired headphones, which admittedly doesn’t hold up in 2019 (streaming and Bluetooth have gotten too good).
Earbuds have small speakers that rest on the ear canal of the wearer. These are usually much less expensive than on-ear and over-ear headphones and reproduce the sound well. However, outside sounds, such as traffic or power tools, can get past the earbuds and interfere with your enjoyment. Earbuds also don't stay in place very well if the wearer is moving much, such as exercising. These also are lower fidelity than on-ear and over-ear headphones.