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I've noticed that many scenes in Vidocq look very cheap, like a cheap TV series. Wikipedia article on Vidocq says:

As the final PAL DVD release shows, the film was shot 25i (interlaced), and only special effects shots were deinterlaced by means of smart field blending (imitating a progressive-type amount of motion blur due to a different shutter speed of progressive modes) during post-production, as the special effects crew obviously was in demand of progressive frames which are easier to process for special effects, before for the final release all normal shots were deinterlaced by means of simple line interpolation instead. The result are video-like appearance of motions in normal shots, and distinctive film-like motions for all effect shots due to the different amount of motion blur resulting from the different deinterlacing methods.

I've read How to get the Cinematic look and why movie video is different from other videos (sports, news, or other), but they don't discuss motion blur.

Why different deinterlacing algorithms make such a difference? Is it really just the sharpness of video picture vs blurred picture for a movie or, in fact, Vidocq has just been filmed cheaply with poor lighting, for example?

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The complete answer to your question, which definitely includes motion blur, but also more than motion blur, is found in this article in The Accidental Scientist. To summarize:

The human vision system has at least two major components that work together in a counter-intuitive way. Persistence of Vision makes it possible for a sequence of images playing at only 16 frames per second to create the illusion of motion. It is not a super high-quality illusion of motion, but the eye sees it as actual motion, not a sequence of images being flashed in rapid succession. Ocular microtremor is a phenomenon that explains why the acuity of the eye is more than 4 times the Rayleigh limit. And its oscillation frequency ranges from 70-105 Hz, and it averages around 84 Hz. A playback frequency 16fps or faster, but 42 fps or less means that the eye both perceives motion and can super-sample the image. Frame rates that are greater than 1/2 the microtremor rate (which for some mean 35 fps-70 fps and for others 53 fps-105 fps), the eye does not get to super-sample the image, and thus feeds the brain an image that fools persistence of vision, but doesn't satisfy the ocular microtremor system. Once the frame rate exceeds the frequency of the microtremor system (120 fps easily does this), the eye sees that as "actual reality". The article argues that the "uncanny valley", between 35fps and 105fps, is why 48 fps movies are not perceived to be 2x better than 24 fps movies, but actually worse.

Where motion blur comes in is that it softens the edges of objects in motion, letting the brain, not the eye, fill in the details. Evidence suggests that the standard formula that shutter speed should be 1/2 of the frame rate (1/48th of a second for 24 fps playback) provides the ideal balance between the brain-based system and the eye-based system to work together.

When a shutter is open for 1/2 the frame rate, it's called a 180 degree shutter (in part because that's how it was created mechanically back in the day). One can shoot with a more open shutter (up to 360 degrees) and one can shoot with a more constrained shutter (144 degrees, 120, 90, 60, 45, etc). Different shutter angles create very different looks in part because it plays with the balance between these two visual systems. The D-Day invasion footage from "Saving Private Ryan" was shot with a narrow shutter angle specifically to set the eye on edge.

All this to say--shutter angle and motion blur are very much a part of the perception of "smoothness" or "sharpness" of the motion. Interlacing is an artifact that mostly just messes things up in the progressive video world in which we now live. The small details on how people get around interlacing artifacts has little to do with understanding the "Cinematic look". The Cinematic look is really based on the deeper principles of shutter angle, frame rate, and of course lighting, lensing, atmospherics, etc. Interlacing either degrades things a lot, or not too much, and while it's strange that somebody would choose two different approaches to interlacing artifacts in a single production, that's less to do with explaining anything deep about the Cinematic look and perhaps more about the discipline (or lack thereof) of the production team.

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