Purchase and Buy are essentially the same word. Purchase is a little more formal than buy.
In addition, buy has usages/meanings that cannot be shared with purchase.
Examples:
You can use 'buy' to mean 'believe':
"I told her a white lie, however I am not sure if she bought it."
In regards to time we can say:
"I need to buy more time!" - which usually means you're running out of time and that you're desperate for more, like to complete a project.
You cannot use purchase in the two previous examples.
Because of these alternate meanings, buy, is used a lot more, is a lot more common then purchase.