Phobia: excessive, irrational fear of a stimulus/event that is distressing to a person so they avoid all possible contact with it, thus interfering with everyday functioning. Examples of 'phobias' include social phobia, agoraphobia (note that these are more GENERALISED)
Specific phobia - also known as simple phobia - an excessive, irrational fear of a SINGLE stimulus that is distressing to a person so they avoid all possible contact with it, thus interfering with everyday functioning. An example of a specific phobia is an animal phobia (e.g. of spiders, or mottephobia).
Thus, specific phobia is a branch of the 'phobia' term.
Phobias interfering with everyday functioning vs. interfering with specific day to day activities is the SAME THING (i.e. day to day = every day).