I think that there are a lot of very subtle issues around the whole British/English thing.

Part of it springs (largely) from the 1970s, when there was a reasonable amount of racial tension in the UK and send-'em-back-where-they-came-from groups, such as the National Front, prided themselves on being "English" and rallied around England's flag, the St George's Cross.

The NF and their like were popularly portrayed as violent idiots and, to some extent, that image has stayed with those who display the flag and describe themselves as English, rather than British. Certainly public displays of either the St George's Cross or the Union Jack (the flag of the UK) carry a whiff of the vulgar with them (except at Royal events and the last night of the proms, of course).

Today I'd say that it's a minor political statement to describe oneself as being English, and it would mostly be seen as either a hankering for the "good old days", when top-flight comedians could fill their prime-time Saturday shows with Paki jokes, or a statement against the European Union.

It's a complex issue, and I'm sure that other Brits will see it differently.