Dear Manasi,
Essentially the 3 tests you mention are very close cousins of each other. Yet, in spite of their closeness, they each serve a distinct purpose.
Psychometric testing is conducted to analyse a person's reflex responses to stimuli, conditions, situations etc... within varying parameters. That is why these tests can be upto 400 questions long with many questions repeating themselves numerous times. But, each time they repeat, they are written with subtle variations from the previous and ensuing instances. This is a deliberate exercise aimed at confusing the human mind into 'remembering' the previous answer. Yet, each instance is the same in its essence. So, the idea is to mainly guage the person's consistency for each category of questions by taking an average of all its instances with subtle variations. In my opinion it is next to impossible to 'learn' how to do these exams. Because, by their very nature, they test reflex responses.
Aptitude tests are designed to specifically categorise people into pre-determined categories that have evolved with decades of psychology research. Their construction is aimed at determining one's proficiency (or lack thereof) in each category. The end goal being to compartmentalise people into their highest average category and ascertain whether one has the 'aptitude' for what ever it is that the test is being administered. Hence, the name 'aptitude test'. Again, scoring is usually a complex relational exercise which is really difficult to learn.
These tests are conducted largely by employers and recruiters to measure and assess individuals against pre-defined benchmarks to ascertain whether or not an individual will be suitable for a particular role, industry, domain etc... That is why they never have a pass or fail. Their only result is 'suitable' or 'not suitable'.
An IQ test on the other hand, is a measure of one's capacity to reason, logic, utilise their brain power etc... Once again, the scores are measured against a pre-defined template to establish one's prowess and capability to deploy their mental faculties. These days, an IQ test is not used widely because of its very generic nature as well as its inability to be customised to fulfil an end. That is where psychometric and aptitude tests have come in and filled the 'gap'.
I hope this helps you.
Kind regards,