And a telecine process of standard 24fps motion picture to video faces the problem of differing frame-rates but in the context discussed here, one would obviously synchronize the frame-rates of the film projector and the video recorder so no artifacts show up. Same with the shutter.
That wouldn't eliminate artifacts as you might expect. Now all my knowledge on this is derived from my years as a Doctor Who fan, and the state of Doctor Who episodes from before 1975 in the BBC archives is a matter of great discussion in the fan community. Everything I'm about to say applies to British TV, basically.
Video doesn't have a 'frame rate' as such. On film you take a discrete image every 1/24th of a second. On video you are effectively scanning the scene with a series of horizontal lines. The crucial difference for our purpose is that standard video scans (for example) all the even numbered lines
then all the odd numbered lines, then back to the even numbered lines. What's more it does this at about twice the rate of a film recording. The result of this is effectively half the lines update every 1/25th of a second, the other half at the same rate but offset by 1/50th of a second. Visually this provides a much more fluid motion than on film, and is similar to a 50 fps film playback. (Recently there was some complaint about The Hobbit because to get the 3D version they recorded at 50 fps to get the effects they needed, and a lot of people felt it looked more like a TV show than a big budget film, mainly due to this difference in how motion looks on film and video). That effectively eliminates one well-known artifact of film recording: motion blur. Even if you sych up the video and film so that one set of lines updates at exactly the same time as the film frame changes, you will simply transfer the film motion blur to the video.
(To complicate matters even more, there was also the difference between stored and supprssed field film transfer, but we won't go into that. Suffice to say that if they used the wrong one the result was that everything that moved on the video produced a double image on the film.)
On top of that there are things that happen on one medium that just don't happen on the other. Take a look at video and film taken of a burning torch, for example. On film you get a lovely image of a flame flickering. On video you tend to get a bright, formless blob, with colour fringing if you are using colour video, and if you move the flame you get a bright formelss blob witha big streak behind it. Bright things bloom and streak on video in a way they don't on film. Transferring from one medium to the other won't eliminate that problem. If you have video of a cleanly burning flame you can bet it was shot on film originally. If you have film of a bright blob you can bet it was shot on video.
Also, consider that when you look at British TV shows up until about the late 80s, as ka9q says, it was common to shoot studio scenes on video and location scenes on film (and some of the more difficult effects sequences as well). When you watch the show on a TV you are actually watching a transmission using a videotape of the entire episode. In other words you are watching a film sequence that has been tranferred to video. The difference is marked.
In the case of Doctor Who it gets even more complex. Taking the old black and white episodes as examples, nearly all of them were shot on 2 inch videotape (405 line PAL originally, 625 line PAL from about 1967). Film sequences were also shot on 16 mm film and then transferred to video tape to be spliced into the episode. The whole episode was then transmitted from 2-inch videotape. For overseas sales a film print was made by pointing a 16 mm film camera at a large sreen. These days, these film prints are all that survives in the archives (assuming the episode survives at all: 106 episodes are still missing entirely).
So, with all the various processes involved there is VHS made from 16 mm film of stuff shot on video. There is VHS made from 16 mm film of stuff shot on 16 mm film then tranferred to video. There is some 35 mm film in there too. When we get to colour episodes there is even more variety, with PAL back-conversions of NTSC conversions of colour PAL video, recolourised episodes made from cleaned up black and white 16 mm film prints of PAL colour material which used an NTSC colour signal to put the colour back, or the recent 'chroma recovery' process. There are manually recoloured black and white episodes. There is also a process called VidFIRE that restroes the video look to film prints by interpolating an intermediate field and interlacing them as on video. This process can't deal with motion blur, however, and so you can even tell the difference between an original video shot and one that has been transferred to film and then reconverted back to the video look. Quite often on the film prints you can also discern the pattern of the projection screen overlaying the action!
In the 90s the BBC tried a 'filmising' process to try and make episodes of TV shows shot on video look like film by effectively removing and deinterlacing alternate fields on the video. The result was terrible, since it couldn't compensate for the different lighting requirements of film and video. These days it has improved greatly, with digital techniques for regrading and so on. Today's Doctor Who is shot on digital video that is processed to give it a more 'filmic' look, and it looks superb.
All in all, the lesson here is that the differences between film and video are significant and not easily removed or obscured by recording on one medium and transferring to another. I can watch all those various examples of TV shows and pretty much tell which was used where. The technology to do it really has only just been developed with digital techniques. With analogue technology I doubt it would have been possible at all, at least not without a substantial research and development program that would have left an evidence trail just as large as the one that details the actual techniques used in Apollo to provide live imagery across the Earth from a small, battery-powered, hand-held video camera on the moon.