Trying to tackle the issue, we need to answer two questions first:
1) Is there a difference between fear, anxiety and phobia?
- It’s a natural response to a threat that can be either perceived or
- Fear is the emotional response to real or perceived imminent threat,
[...]. (2, citing DSM-5)
- Anxiety is a word we use for some types of fear that are usually to
do with the thought of a threat or something going wrong in the
future, rather than right now. (1)
- [A]nxiety is anticipation of future threat. (2, citing DSM-5)
- A phobia is an extreme fear of a particular animal, thing, place or
situation. People with phobias have an overwhelming need to avoid any
contact with the specific cause of the anxiety or fear. (1)
- Phobias are anxiety disorders involving fear or anxiety that is
excessive or out of proportion to the object or situation. (2)
(1) How to overcome fear and anxiety, Mental Health Foundation
(2) Classifying Anxiety: Understanding the Difference Between Fear and Phobia, Grace College Online
So my answer is yes.
2) Is the difference important from a parenting perspective?
When determining which tags to keep / create, this is also an important question to ask and answer. Imagine questions about a child's spending the whole day on the Internet - it would not make sense to have individual tags for each and every site (the question may even be answerable if we don't distinguish, for example, between forum activity or playing browser games).
Another point to consider is whether the tags are used the way they should. This is usually a minor point as incorrectly tagged questions can be edited. However, it could turn out that one tag is always incorrectly used and that there actually are no questions for this tag.
Also keep in mind that requests for medical advice are off topic. For example, we have the tag injury for parenting questions about injuries - do we need to differentiate further? Not really, as how to treat them is mostly out of our scope. Further explanation can be given in the body of the question instead.
Too many questions are tagged fears and/or anxiety to too much into detail. While some questions are mistagged, there are enough questions for both tags to make keeping them sensible.
So my answer is yes.
Only 3 questions are tagged phobias:
What are good approaches to lessen fear of insects in a child?
How to help a child overcome fear of heights
How to help a child with a long-term weather phobia?
They all describe a phobia, so the tag seems appropriate. On the other hand, questions 1 and 2 could be tagged fears, while question 3 might be anxiety instead. I feel that in this case, like in the injury example above, we could do without the more specific tag. The body of the question could then be used to specify the problem.
So my answer is rather no.
I suggest to keep fears and anxiety. If you find mistagged questions, edit them or suggest an edit. Improve the tag wiki excerpts, especially the anxiety one to give a brief description of what anxiety is. Maybe also add a reference to the other tag to guide users when to use which one.
I suggest to burninate phobias. Do not make it a synonym of either fears or anxiety, since a phobia can be based on either one. Instead, users should use the appropriate tag fears or anxiety and then describe the phobia further in the body of the question. Unless we suddenly get many more on topic questions about phobias specifically, we can do without phobias for the moment. If you have too few questions, the tag becomes less useful for searching.