Adware is a type of malware that displays advertisement and pop-ups on the screen. Hence the name Adware
This can result in performance issues as certain amount of processing power or computer resources are utilized in fetching these advertisements and displaying on the computer screen.
They are usually installed while installing malicious softwares.They generally don't send the victim's data to the attackers. Their main aim is to promote their product (like thier own adware removal product that creates a backdoor on your computer) or the product of 3rd party companies.
Spyware is a malicious software that spies on you. Hence the term "SPY" prepended to "WARE".
It can do a lot of things besides spying on the sites that you visit.
It can even inject some malicious code on the sites that you visit and cause your session tokens,IDs to be sent to the attackers and if this wasn't enough it may even wait for you to input some data (like credentials) so that it can send it to the attackers.
It caputres the victim's surfing habits, and can even install its own keylogger.
NOTE:The attackers know that the victims look for free spyware removal tools online and therefore create their own versions of it. When these tools are installed, the spware gets automatically installed on to the computer.