After Picard asks Dr Crusher to report to the bridge to meet with the Aldeans, after mentioning Starfleet Regulation 6.57, he walks away from Data who is at the helm. When the scene goes back to Data, Picard's hand is clearly visible in the short, although he already clearly walked away. The shot of Data is taken from just a few seconds earlier.
The Aldeans abduct seven children from the Enterprise. However, in the following scene, Harry is absent, meaning that there are only six children on Aldea. Harry appears without explanation in the next scene set on the planet.
When Riker, Troi, and Crusher are beamed down to the planet, Crusher and Troi are leaning on the bridge tactical console, yet when they arrive on the planet, they are standing upright. Similarly, when the three are beamed back to the Enterprise, they depart while seated but arrive standing up.
Before and after Picard introduces himself to Rashella, a lot of changes take place. Worf is telling that the Enterprise is being scanned, but he isn't looking at the screen. After that, Lieutenant Yar is opening hailing frequencies. After Rashella introduced herself, Yar is suddenly leaning on the panel. Tasha's left arm in the same coordinate of the right commanding chair. In the next shot, you can see Dr. Crusher quickly crosses her arms between shots and Tasha standing more to the right.
Data explains that the Aldeans cloak works by bending light rays around the planet. This would mean that no light or other electromagnetic energy could reach the planet's surface, leaving it in complete darkness and completely frozen. It would be so cold even the planet's atmosphere would freeze and fall to the ground like snow. This would also not hide the planet's gravitational field, which would give its location and clearly as if there were no cloak at all.
Granted this was done to include one of the principal actors, but surely there were other "special" teenagers who would be transported to the planet in addition to Wesley. The Aldeans cater to the children's gifts, and there would be a great many more aboard the Enterprise who were Wesley's age and had unique talents.
Seven children alone would do nothing for thousands of barren Aldeans. If the goal is for reproduction they have been rather stupid stealing 4 girls and 3 boys. Even if the Aldeans can repair genetic defects, 3 males and 4 females would mean most of the next generation would be at least half siblings. The following generation would ALL be first cousins. The lack of genetic diversity would make it difficult to create a viable population. The inevitable inbreeding would lead the emergence of many recessive disorders. However, the Aldeans have advanced technology, which would include gene editing to deal with this. (On Earth, we already have it in the early 21st century, albeit in its infancy.)
In addition, none of the children, except Wesley, is old enough to have children. This means waiting several years before the first baby could possibly be born. The Aldeans presumably expected the Enterprise to be only the first of many ships they could ensnare, taking a few people from each one.
When the Enterprise is hit by the Aldea repulsor beam and is spinning out of control, the main bridge alert indicator behind Riker (and beside Worf) is flashing at Red Alert (at the same time as the audible klaxon) while the other alert indicator behind Dr Crusher is still indicating Yellow Alert.
When Wesley beams into the Custodian's room, the picture shifts slightly, thus revealing the cut.
Right after the Aldeans beam off the Enterprise bridge, a tear or seam in the carpet is visible.
At the beginning you can see a lot of small white markers for actors' positions on the floor of the Bridge.
Near the beginning of the episode when Riker is telling Tasha about the Aldeans, his lips are saying something else, and the words Aldea and Aldeans are dubbed in.
The intruder alarm goes off well before the Aldeans beam aboard.
At the start of the episode when Riker goes into the turbolift after his collision with the boy, you can see the foot of some sound equipment sticking out. This has been digitally removed in the HD update.
The central conflict of this episode is that humans would never sell children. However, it would not be necessary to sell children. Even in the Star Trek universe there would be orphaned children in need of adoption. It is even a major plot point in the series that Worf was adopted, so the crew should have been aware that was an option he could have offered up front and avoid any conflict.
Dr. Crusher expresses concern that the Aldeans have not been through any kind of decontamination. While they explain that the Federation transporters would not have worked, they do not assure anyone that their teleporter has a decontamination protocol.
The Aldeans offer Picard information on "parts of the galaxy you do not yet know exist." Since Aldea has been hidden behind a cloaking shield for millennia, there is no way for its people to know if the information they have to offer is still current and valid. Their data may be wildly out of date.
When Harry's father is telling Harry that he "still has to take calculus," the shadow of the boom mic is visible in the upper right (on the door frame).