Is there a word to describe a person who always buys what's new?

I could probably suggest "fool" and be accurate enough, but I suspect it may put some noses out of joint if the wrong people find this answer in their search results. (Really people, It. Is. A. Phone. Nobody who has a sense of priorities cares.)

I'm not aware of a single word that's commonly in use. However the most common expression that I've come across is Early Adopter.

A person who is quick to follow latest trends is, well, trendy

A person who is very fashionable or up to date:

Taiwan’s well-heeled young trendies

Although this doesn't quite capture that he does it to "show off and feel like he is the coolest person in the world."

For that, try

swanky

fashionable and expensive in a way that is intended to impress people

a swanky new hotel

PS - note that swanky is approving. You sound like you need a word that's disapproving.

Referring specifically to technology products as per you example, a slang expression is Technofreak:

• Somewhat like technogeek, a person who is obsessed with the latest technology, and everything techno-y, (music, clothing, etc), while still being able to straddle the line between nerd/geek and normal person, with a reasonalbe social life. They are also extremely adept at new technologies and are usually able to complete really complex technological things without becoming too unbalanced. Usually has a nostalgia for the 90's rave scene, even if they weren't involved in it.

(Urban Dictionary)

This would be an "early adopter".

In the diffusion of innovation theory, the minority group (comprising about 14 percent) of population which, after innovators, is first to try new ideas, processes, goods and services. Early adopters generally rely on their on intuition and vision, choose carefully, and have above-average education level. For any new product to be successful, it must attract innovators and early adopters, so that its acceptance or 'diffusion' moves on to early majority, late majority, and then on to laggards.

Read more: http://www.businessdictionary.com/definition/early-adopters.html

Modish is a possible word.

modish ˈməʊdɪʃ/Submit adjective: modish conforming to or following what is currently popular and fashionable.(Ref)

I believe a good word for a person who likes or buys new things all the time is a:

Neophilia / (ˌniːəʊˈfɪlɪə) / noun a tendency to like anything new; love of novelty