ŷ is the prediction you get out of the formula, β̂ is the coefficient(s) on the variables in the formula.
So, take the simplest possible model: You think that your dependent variable is linear in the dependent variable, and that the line passes through (0,0). That gives us a simplified version of the classic equation for a line, where we say that y is equal to x times some constant that indicates the slope of the line:
y = mx
In machine learning (or statistical inference), the variable conventions are slightly different, so it would look more like:
y = βx
And then, since this is machine learning (or statistical inference, or whatever), we don't know the true equation, we're just working with an estimate we've inferred. We stick hats over the estimated values to make it clear that they're estimates:
ŷ = β̂x
What I was describing, then, is which side of the formula you're focusing on. Traditionally, statistics has been all about scientific inference, and trying to figure out and characterize causal relationships. So you want to know "If I change x in some way, how will that affect the outcome, all else being equal?" β̂ is your tool for teasing those things apart. Machine learning is typically cast as being more about making predictions from observational data. ŷ is just the formal notation for describing those predictions.
It's pretty common in machine learning to say, "We are working with observational data, we're not worried about causality, so we'll only pay attention to the predictions." There can be value to that approach, but only to the extent that you really don't care about causality. Which. . . surprisingly often, β̂ matters, even when the people who built the model were trying to ignore it. It becomes especially important when you're using the model to make decisions. For example, when a company is using a machine learning model to try and identify candidates for a position they're trying to fill, they should absolutely be paying attention to what the model thinks about people who liked Lane Bryant on Facebook, and how that compares to people who liked Jos A Bank, and whether they really want things like that to be playing into how they make hiring decisions.
