A number applies to a class, rather than to an individual. The number three doesn't apply to each individual vertex of a triangle, rather the number three applies to the class of all of the vertices of a given triangle. Even if there is only one object, the number applies to the class of those objects, not to the individual. When you say there is exactly one current president of the US, the number one does not apply to the individual named "Joe Biden"; it applies to the class expressed by "current president of the US".
Many (probably most) philosophers would claim that by contrast, existence is something that applies to an individual. For example, George Washington no longer exists. This is a statement about the individual George Washington. However, there is a problem with this account, and that is that it is hard to assign a semantics to a sentence like that. If George Washington doesn't exist, then what does the name "George Washington" refer to? If it refers to nothing, then the sentence reduces to "nothing doesn't exist", which is a bit odd.
To deal with this and other problems of the view that existence is a feature of individuals, some philosophers claim that existence is just like number: it applies to classes rather than to individuals. For example, when you say that are no unicorns, you are saying that the class of unicorns is empty.
This doctrine has to account for existence statements of the form "Joe Biden exists" which seems to be about an individual named "Joe Biden". One way to deal with this is to say that the sentence is really about the concept of Joe Biden rather than the individual Joe Biden. The concept of Joe Biden has a class of individuals that instantiate that concept. The class has one member. Thus, to say that Joe Biden exists is to say that there is something that instantiates the concept of being Joe Biden. This works equally well with George Washington who does not (currently) exist. To say that George Washington does not exist is to say that the concept of being George Washington is not instantiated.
Of course, there are many arguments on both sides of the issue, which I won't go in to here. However, with that background, here is the short answer to your question:
If you believe that existence is a property of individuals, then number and existence are not the same thing because number applies to a class and existence applies to an individual. Number is about a class of which the individual is a member, existence is something that the individual has. If you believe that existence is about classes rather than individuals, then, yes, saying that there are zero unicorns is pretty much the same as saying that unicorns do not exist, in the same way that saying a set has zero members is the same as saying the set is empty.