What chemicals/substances exist in green capsicum (peppers) that do not exist in red capsicum?
Your limited understanding is indeed correct - the different colours indicate different stages of maturation, even though different colour cultivars exist (e.g. some that remain green even if fully ripe). 

It is known that the different maturation states have different vitamin and/or mineral concentrations (see here or here). This indicates that different biochemical pathways are active and that therefore the enzymatic set-up is variable during pepper maturation (which could have an effect on tolerability as it might also affect other plant components).

Moreover, you are right that the green colour is - at least partially - caused by chlorophyll but also non-green bell peppers often have chlorophyll (Lancaster et al., 1997) but that probably is not the cause of your friend's responses.

More likely the cause is 3-isobutyl-2-methoxypyrazine (IBMP) which gives the characteristic (green) pepper smell and has highest concentrations in green peppers (that's why it is called green pepper smell). It has one of the most intense odours that can be detected at very low concentrations, even within the complex odours of wine (Pickering et al., 2007), according to Roth, 2014 perhaps even below PPT concentrations. 

So how does this fit together? Pickering et al., 2007 also suggest that one should determine 'consumer rejection thresholds' for methoxypyrazines, indicating that it might be repellent to some people even in low concentrations of 1-2 ng per litre. This could explain your friend's aversion but evidence for the allergy component is dubious. 

I did find one paper (Jensen-Jarolim et al., 1998)  about allergen expression in peppers (birch allergy like). However, no green-specific allergens are reported, and it even seems like green peppers would be a better choice to avoid these allergens.