The word der Bildschirm means computer monitor. I observe that there is another word der Monitor which also means computer monitor. I want to know, is there any difference between these two words?
When it comes to computers, no, there is not. "Monitor" and "Bildschirm" mean the same thing. Much like "monitor" and "screen" do in English.
For reference in German, this is how the Duden defines "Monitor" in the context applicable here:
Bildschirm eines Personal Computers o. Ä.
If you want to nitpick (which you shouldn't), the "Bildschirm" is only the "screen that displays the image". In monitors of yore, i.e. cathod-ray tubes (Kathodenstrahl-Röhren), it designated the phosphorescent surface that the electron beam would hit, thus making the image appear. The "monitor"/"Monitor" (in English/German) would then be the integral "electronic device with a screen used for display", as Merriam-Webster defines it.
But again, the two words are used interchangeably. Not just because the days of cathode-ray tubes are long gone. Also because people don't usually dwell on that technical distinction.
The word "Bildschirm" means the screen, either of a computer monitor, a TV or other similar device, although it would not be used for small displays but mainly for bigger ones that used to have a front glass with a CRT. Note "Bildschirm" is just the front that displays the image, not the whole CRT. There days it is usually used for LCD or similar displays, as CRT are no longer in general use.
The word "Monitor" applies to a device that includes a screen ("Bildschirm") but doesn't contain a TV receiver. It may be used for computers or surveillance cameras or similar applications.
Whether a device includes a TV receiver is important because there are fees to pay for devices capable of receiving TV signals, whether they are used for that purpose or not, that are not in private use.
If it is about computers, there is no difference at all, only speaker's preference.
For general use, Bildschirm has a slightly broader range than Monitor, for example the screen of a TV (especially an old CRT one) would be called Bildschirm but not Monitor. Yet, to be precise, this is true for an entertainment TV, while a screen for surveillance (images of security cameras) would be called Monitor even at times when it was technically not different from a TV set.
As stated in other answers: Bildschirm focuses more on the displaying property, while Monitor is more appropriate for a separate stand-alone device with power supply and connection plugs. (I can't remember having seen a laptop advertised as mit Monitor). If the size is more on the smallish side, one would more likely chose one from Display or Anzeige.
Since Monitor has numerous additional meanings (one can monitor noise, carbon dioxide, amount of rain), I would always prefer Bildschirm.
Especially frequent is the word monitor for some high quality speakers in a sound studio.
From the ancient history a monitor earlier was something without UHF decoder, which distinguished it from a TV set, just able to produce some digitally applied signal without the quality loss via the conversion of the appropriate modulator (as found in ancient home computers).