Upon leaving the Horizon, Travis tells Paul that he will ask Lt Reed to help remove the locating beacon attached to the ship's hull by the pirates. However, when he gets back to the Enterprise, he simply tells Archer that everything went well during his visit and the Enterprise departs immediately.
When Horizon fires upon the pirate ship and disables it's engines, it slows and comes to a stop. In space this wouldn't happen. The ship would keep moving - even without continued thrust from the engines - until something stopped it.
When Captain Archer goes to check on Ensign Mayweather in the zero-gravity section of the ship, Mayweather is visibly upset and at least one tear falls down off his face.
When Captain Archer goes to see Mayweather and he is in zero gravity, the captain's hair is not out of place. His hair should be at least standing up.
When Captain Archer and Mayweather are using the zero gravity area of the ship, you can see rips in the hip areas of their uniforms. This is possibly because of the wire harness they would be wearing to give the effect of weightlessness.
Travis deliberately hides the Horizon's experience with the space pirates from Archer when the latter mentioned seeing hull damage on the ship, likely to show that he thinks his old cargo ship is fine on its own. But a Starfleet officer would certainly report this right away, both so the Enterprise could be on the lookout for the enemy aliens and so Starfleet could warn other cargo ships.
When Travis enters his old room, the chess board is set up with black in lower right.