What problems did the Puritans have with the Church of England?

If you return to the original Lutheran Reformation of the
Roman Catholic Church, you find that implicit in Martin Luther's original articles of
faith--the priesthood of all believers, the authority of the Scriptures, and the
doctrine of salvation by faith alone--is the rejection of the ecclesiastical hierarchy
(and bureaucracy) that had evolved since Christ gave the keys to the Kingdom of Heaven
to Saint Peter.  Luther's famous rejection of Pope Leo's indulgences led inevitably to
an outright denial that the Roman Catholic practice of Christianity was still consistent
with the original precepts of Christian faith.  Of course, because he had to maintain
good relations with Frederick, Elector of Saxony, for his very survival, there was no
way Luther could continue that line of thought to its logical conclusion that a
priesthood of believers meant a community of equal brothers, which would certainly
threaten the social and political order of the
day.

However, in England, where the Reformation had been
rather cynically imported by Henry VIII to justify himself as Head of Church and State
and to relegate the Pope to no more than the Bishop of Rome, many churchmen did pursue
Luther's original train of thought and called into question whether aristocracy was
essentially incompatible with Christian practice in its purest form.  (This was not a
new development, by the way.  John Ball had raised the same questions in the reign of
Richard II, preaching in support of the Peasants' Revolt against the very notion of the
upper and lower classes.  Jan Hus had made a similar, although less radical, protest in
Bohemia in the early 15th century.  Needless to say, those ideas were easy to crush when
they were merely local; with the printing press, however, ideas took flight and the call
for reform was everywhere.)  Puritanism demanded that the English (Anglican) Catholic
Church extend its reforms of Roman Catholicism and return to the simple community of
brethren and believers that they read about in the description of the early Christian
Church in the Book of Acts.

Needless to say, not only did
the Puritans have a problem with the Church of England, but the Anglican
hierarchy--which was in most respects quite similar in practice and structure to the
Roman Church--had a fundamental problem with the Puritan vision of reformation.  When
Henry was enforcing his will on both church and state, he famously sent both Roman
Catholic and Puritan clergymen to the stake in pairs, chained to the same pyre, for the
offense of not subscribing to the middle ground of reform that he dictated for his
people.