The Battle Demi-Company is the center-piece of the Gladius Strike Force, and the basis of most Space Marine armies in the fluff. In theory, it's exactly what it says on the tin; half of a Full Battle Company. It requires a Captain or Chaplain to command it, 3 Tactical squads, and Assault squad, and a Devastator squad. On top of this, you can add a Command squad and any type of Dreadnought squadron.
Digging even deeper into the fluff, the Assault squad can also be fielded as a Bike squad, Attack Bike squad, Land Speeders, or Assault Centurions. All of these squads are filled out by standard Assault Marines in the background. The same applies to Devastators, who can be fielded as Centurion Devastators as well.
If you field all of these squads, you get several nice special rules. First, all of the squads gain Objective Secured. This rule doesn't seem too exciting because Troops reliably get access to it via the Combined Arms Detachment, but you get to spread it a bit wider in the Demi-Company. Beyond Tactical squads, it can apply to very mobile squads like Assault squads and Land Speeders, as well as Devastator squads, excellent for holding backfield objectives. In my mind, the biggest advantage is that your Chapter Master with Shield Eternal, leading his tremendously dangerous and durable Command squad, is Objective Secured. These guys are almost guaranteed to last until the end of the game, and will be quite difficult to forcibly remove from an objective.
Second, the Demi-Company gets access to single use of the Tactical Combat Doctrine over the course of the game. It only affects squads in Demi-Companies, but it's a great ability to be able to pull out for an important turn.
Competitively, the big advantage for the Demi-Company comes when you take two together within a Gladius Strike Force to form a full Battle Company. When you do so, all of the Dedicated Transports taken by the squads are free (though you still have to pay for any upgrades). If you take Razorbacks for all of the possible squads, you could get 660 free points in your army. Obviously, that's a very powerful option in a competitive environment. Unfortunately, using the Demi-Company in this way isn't at all appropriate for friendly or fluffy games.
My biggest problem with the Demi-Company isn't what it requires, but how people field it. Minimum sized Marine squads are the norm, to both minimize the cost of the Demi-Company and maximize the usefulness of Razorbacks. This is all well and good, and Marines are sometimes fielded this way in the background. The problems comes with the Battle Company, which is supposed to contain roughly 100 Marines. The most competitive way to field a Battle Company contains only around 40 Marines, less than what's supposed to be in a Demi-Company. If you're playing in a cut throat tournament, then you should definitely consider this style army. If you're playing more relaxed games and you want your normal opponents to keep playing you, I'd suggest trying to be a bit closer to the fluff.
You can still field a Full Battle Company in a fluffy manner, and you certainly should. It's a great way to represent the way Space Marines are supposed to fight. Fortunately, all you really need to do to be a bit friendlier with the Battle Company is take some full sized squads, preferably with Rhinos or Drop Pods rather than taking all Razorbacks. This not only gets you closer to the fluff but also reduces the overall effect of the free transports on your army. I'm personally looking forward to fielding a Battle Company with a tremendous number of Tactical Marines in Rhinos, with Razorbacks for the Devastator squads. I love playing hordes of Marines, and that list seems like it might actually play well. Alternatively, the Battle Company could be a very strong Drop Pod list that would still be fun to play against.
I guess my point throughout all of this is that the Demi-Company is a good formation, and the Battle Company has the potential to be exceptionally strong. If you're not playing in a really competitive environment, you may still want to field a Battle Company because it's so fitting of the background (seriously, how many Marine players do you know that are building a Full Company? I did for my Blood Angels, and it looks like I'm on the same track with my Executioners). You should certainly feel comfortable fielding a Battle Company in such a case, and no one should be in a position to give a hard time as long as you're putting at least 60 Marines on the table.