Two major reasons. One, the obvious difference between the two- Irish immigrants tended to form tightly knit social communities, partially due to being forced out of/kept away from other communities and partially for their own protection/homesickness/desire for social cohesion. This made it so that they largely retained their distinctive characteristics- accents, religion, diet (in fact, there still exist almost entirely Irish communities in large industrial cities such as New York or Boston). Two, the Irish have a staggering national hatred of the English (to GREATLY simplify several hundred years of political upheaval), which makes the likelihood of an 18th or 19th century Irish family choosing to pretend to be English very little.

The simple, but unsatisfactory, answer to this is that we probably don't know how many Irish emigrants tried to pass as English.

I'm not aware of any academic literature on the subject, but it seems unlikely that nobody would have tried. Beyond the obvious pride in where one came from, some people would undoubtedly have seen that the English were a step above the Irish on the social ladder and tried to take advantage of it.

The practical obstacles, more than any sort of anti-English or religious feeling, would be more important I think. Bearing in mind that the majority of Irish emigrants would be native Irish-speakers, most of whom would have little formal education, it's unlikely that they'd have had the ability to pose as English.

Of course, not all emigrants were economic migrants. There was a proportion of the reasonably wealthy and educated classes from Dublin - who would be native English speakers - who sought a better life in America. It may well have suited them to pass themselves off as English.