'She is arriving on Monday' means that the person is travelling from another place, perhaps another city, country, or continent. She will travel on Monday, or - if it's a very long journey - her journey will end on Monday. 'Coming' can be used to mean the same thing. But it doesn't necessarily involve travel. For example, if you are talking about the lady who comes to your home to do the housework, you could also say 'She's coming on Monday'. You cannot use 'arrive' here, because you are not talking about travel - you are simply saying that she is going to come. Also, note the spelling. There is only one letter 'm' in 'coming'. If you have a word ending in a consonant and then an 'e' (like 'come', 'give', 'drive'), you just take off the letter 'e' and then add 'ing' - coming, giving, driving.

There is not much difference, except that using the verb "arrive" implies distant travel.