Handle :
The Becker is swelled and rather smooth while the Cold Steel is more uniform and aggressive. There are people who prefer either version, depending on the level of security you want, the level of abrasion you can tolerate and the shape of your hand. It is something you have to see for yourself. The Becker however is far more durable in regards to impacts, wear, abrasion and fire.
Blade primary grind :
The Becker is sabre-flat and the Cold Steel full-flat. The Becker makes a better wedge/splitter with a higher edge durablity while the Cold Steel is a higher performance cutting tool. This assumes the edge bevels are similar, and from my experience they are within a few degrees of each other, the Becker being a little more obtuse so again less cutting ability more splitting and more durable. The Cold Steels come with a sharper edge NIB from my experience (small sample).
Point :
The Trailmaster has a much thinner point so you have higher penetraton but lower strength.
Edge geometry :
The Brute has a significant curvature which enhances slicing ability and any draw cutting in general but hampers wide push cutting. The Trailmasters straight edge will make sharpening easier, especially if you are not free handing it.
Steel :
Depending on who you ask and when, the steels may be identical or at least very similar so no great difference in this area.
Coating :
I had a couple of Cold Steel black coats come off easily, the Becker rough coat is more durable, however I had a bad experience with that once as well (Machax).
Sheaths :
The Beckers have decent Kydex sheaths, the Cold Steel has really low grade ones last I checked (a couple of years ago), they may have upgraded since then.
-Cliff
