"Arrogant" vs. "conceited"

Solution 1:

From Merriam-Webster:

Arrogance - an attitude of superiority manifested in an overbearing manner.

Conceit - excessive appreciation of one's own worth or virtue.

Clearly the two words have considerable overlap in meaning in some contexts - though we never say, for example, "He conceitedly elbowed her aside", or "Describing terrorists as 'freedom fighters' is an arrogance that doesn't work", so they're certainly not complete synonyms.

For OP's purposes it may suffice to note that on average arrogance is more "external" - how a person acts towards others, and conceit is more "internal" - how highly he rates himself.

Solution 2:

An arrogant person will boast about how wonderful they are. A conceited person will just assume that you know how fantastic they are, and if you don't, you are probably not worth talking to.

A person who is both will assume you know how wonderful they were yesterday, but will have to tell you how phenomenal they are today.