People use the two terms interchangeably, and it causes real budget problems. The distinction is not academic - it is the difference between a project measured in weeks and one measured in months.
The practical difference
A website presents information. Content changes occasionally, every visitor sees broadly the same thing, and the hard problems are design, structure and search visibility.
A web application does work. Users log in, data changes as a result of what they do, and the system has to enforce rules about who may do what. The hard problems become data modelling, permissions, reliability and security.
A quick test
Your project is probably an application if any of these are true:
- Users need accounts
- Different people should see different data
- Something must be saved and retrieved later
- There are rules about who can change what
- It has to stay in step with another system
Why it matters to your budget
An application carries costs a website does not: authentication, testing, error monitoring, backups, and ongoing maintenance because the thing is genuinely in use. Quoting an application at website prices is how projects end up abandoned halfway.
It also cuts the other way. Plenty of businesses are sold an application when a well-structured website and one integration would have done the job for a fraction of the cost.