Blog is more like a journal/diary one keeps, it's something others can read and comment on, they do not edit the articles posted. You can set up blogs so it's open to everyone or is available to only a closed group. It's more appropriate for personal use.
A wiki on the other hand is like a workspace where articles can be published collaboratively. So for instance if a you want to define a lesson plan with some colleagues, you can create a page in wiki which you and others can see and edit. So many a part of lesson plan is to be handled by spanish teacher and another one by english teacher, they both can work on same plan document. Wiki keeps all the versions so it's easy to go back, and with a small learning curve makes it very easy to create and link pages together. Like wikipedia ofcourse.
In my opinion wiki is more likely to work in school/classroom setup. But there are also specialized software available that allow one to create classrooms, and post announcements, events , homework etc under those for parents and students to see.
Wikis are meant to be used as tools to co-create documents. Blogs are meant to be used as a publishing tool of (relatively) short texts.
If you are interested in publishing short ideas that you think are ready for public consumption, a blog is the way to go. If you are trying to co-create a document with collaborators, wikis will be more efficient.
To make it complicated, you need to know that several persons can work on the same blog, making it a team-blog. Also a Wiki might be used to publish and discuss ideas...
I've found the Common Craft videos: "wikis in plain English" and "blogs in plain English" to be really helpful when explaining this difference to folks. You can do a YouTube search for Common Craft or "in plain English" and you'll see a list of options.