Technically, there does not have to be any difference at all. However, semantically, “analysis” might suggest something more scientific with higher precision and better performance than a simple “monitor” which possibly suggests a more indicative output where precision is of less importance.
With that said, the two words “analyser” and “monitor” are often used interchangeably. By example, you can find multi-component “gas analysers” which are applied to what traditionally is called air quality monitoring (AQM) respectively continuous emissions monitoring (CEM). It is therefore natural to also refer to them as gas monitors. However, the expected measurement precision within these applications fields certainly require “scientific” levels of precision and accuracy in the measurements, suggesting the need of an “analyser”. Likewise, and “oxygen monitor” can perform an analysis of the oxygen content in a gas and can report the oxygen concentration with high precision.
It should be noted that there is at least one class of gas monitors that are of more indicative nature: sensors for indoor climate and ventilation control. For those applications, it is usually enough to know if for example the CO2 content in the air is “high” or “low”, and control ventilation fans accordingly. Such monitors are straightforward and cheap and do a good job in such applications, but they do not have good enough performance characteristics for true AQM or CEM applications.
In any case, whether you buy a “gas analyser” or a “gas monitor” you normally get what you pay for!