The big issue I see overall is a struggle to make contrast work but a hesitancy to actually push the contrast to a readable state. All the semi-transparent rectangles behind information make for a very unclear business card.
Be very careful when you find yourself wanting to put outer glows and drop shadows on text. This is sure sign that there's a readability issue. There really should rarely be a need to separate text from a background with a glow or shadow. When there is, it's an indicator that the contrast between the text and background is too low. For a business card, where conveying information is the paramount issue, low contrast is just bad design. I notice on every single line of text, you've got a glow in addition to the background rectangle.
The logo is just horrible - in both design and name. (Seriously? Putting "bored" with a question mark in your company name and logo? Bad idea) But I don't know how much you had to do with that specifically. Typically clients can supply the logo. If that's the case, you're kind of stuck with it. Unless you try and up-sell them a better logo.
For the front of the card, the frayed scrap of paracord is adding nothing to the card. Honestly, it looks like a piece of trash on a black background. It's not dynamic or intriguing in any way. I'd remove it entirely.
My personal pet peeve.... the social media stuff. Social media is a great marketing tool, but the urls for every single one of them are really not needed. If you are a user of any of the social media platforms you know that simply searching a name will find a company. So really, all you need at most are the icons and perhaps something like "Find BoredParacord on..." once. There's no need to have the primary domains, the icons take care of that. Truth is.... today there's little added benefit to the icons or listings at all. No one ever listed "Yellow Pages: 1.555.555.5555" as a phone number... people know where to find companies if interested.
In addition, it's also a pet peeve of mine to list all the social media URLs but not have a listing for your own domain?? I, personally, don't place much stature on any company which only has a facebook or twitter page. If they don't have their own domain, they aren't serious. Now, the lack of the specific domain may have been an oversight on your part (it does exist). But it should always be part of the information on any company stationary. (I also feel lack of any mailing address has the same connotation - we're not a real business - but that's my opinion.)
For the back of the card, I'd agree with @DA01, the grenade is the most compelling thing about either side of the card. I'd move it to the front of the card. It also appears as though you "squished" the photograph vertically to make it fit the card.... don't do that! If you need to reduce, reduce proportionately. In addition, after a quick Google search, there are better images to use for it. For example....
Assuming you have permission from Greg Pastore to use the image
That yellow highlight is a fantastic element to play off. Set the type in the same bright yellow on a fatigue-green background and it'll pop for certain.
(Note there's also a grenade with a hot pink highlight which would work wonderful as well, but it may be a bit too feminine for some of the guys in the company.)
I'd also suggest not using the back of a card unless it's specifically designed as needed. For example, simply placing the logo and the paracord grenade on the front, then have all information on the back (without any photos).
Take the colors from the photographs you use... the yellow, greens, whites, and blacks. The reds and blues you used are not in harmony with the photography you used. The company doesn't appear to be set on any specific branding colors (they are all over the map). But as a designer, it's best to narrow that focus down to colors which are harmonious and solid when conveying information. You don't want 5 different shades of "blue". It tends to create chaos for the viewer.
A very rough mock-up with some random QR code on it....
Additional about the "contest"......
"Contests" are absolutely horrible ways to try and get work or exposure for a couple reasons. 1) There's never any feedback, as you've seen. There's no telling what the client did or did not like about your design. There's no way to know if you were "close" but just got edged out. 2) You're working for free, even if you do "win". The prizes for most of these "contests" amount to slave labor wages at best. If you want to be taken seriously, you need to avoid the "hey everyone work and we'll pay a winner 1/8th of what professionals want to charge!" avenues.
All that being posted... look at the winner.... what did that person do that you didn't?