I would not like to see a discussion started here with the opening question phrased as it is. Both appellations used are based on assumptions regarding identity which are not complementary in any meaningful sense and therefore what is presented is a false dichotomy. The evidence of this is in the conduct of certain individuals on the thread to which you have linked, one that was correctly closed by a moderator since the flaw in the original premise of the discussion simply opened the door to any lunatic or ignorant interpretation, and enough people duly obliged in that respect.
Your link to your own thread from the old BBC board reveals much the same - the conversation quite quickly derailed into verbose but meaningless interpolations by some contributors and totally irrelevant ones by others.
A discussion concerning the fostering of scientific knowledge and its contingent ideologies and philosophies within areas governed by Muslim rulers at a time when such systematic protection of lore was noticeable by its absence elsewhere is almost guaranteed to invite participants to take exaggerated stances concerning each premise. The truth, as ever, is less facilitating to such extreme views, however seductive they might appear to be to certain individuals to adopt them. Nurturing of scientific debate and exploration was not uniform at the height of the Caliphates throughout the regions they controlled. Some areas under their control which had long valued scientific and philosophical pursuits within their culture continued in that vein with local patronage and approval from the authorities. Some which did not have that tradition became comparable due to similar support (Fez, in Morocco, for example). Others however lost such status under Muslim rule.
By the same score "Europe" during the same period developed just as unevenly. Divorced from several ancient seats of learning with the final collapse of Roman hegemony and the subsequent isolationism and then shrinking of the Eastern Empire, it is not surprising that the development of intellectual endeavours took a dive in some places and underwent a fundamental transformation in character and scope in others. But there are many examples of resurgence and even continuity of state-protected areas where learning and discovery could be undertaken to a point where they could still be reckoned as endemic features of the local culture. They might have been few, but they were no less important for that - Ravenna, for example, springs to mind immediately but there are less well appreciated centres from Cashel in Ireland to Krakow in Poland (and many places in between) where education was taken as seriously as in any Caliphate equivalent.
When the original question posits a "decline" it invites us to suppose that "Islam" somehow took its eye off the ball with regard to the pursuit of knowledge at one point while "Europe" did not. This would be to ignore the manner in which knowledge itself was regarded and imparted in the period up to the 11th century - in both Muslim and Christian societies a pursuit inextricably linked to religious instruction. It wasn't until the establishment of an institution in Bologna which intentionally set itself aside from overt religious affiliation (and first coined the term "university" on that basis) that one could even begin to imagine a pursuit of knowledge for its own sake, at least in Europe. Byzantium and the great Caliphates of the same period resisted this development strongly, indeed one could argue that the fractious and highly divided politics of the rest of Europe was ultimately what allowed the development to take place at all.
This is the only sense in which I can imagine a "rise" or "decline" being used to describe the relative fortunes of science without and within the Muslim territories. But yet it is contingent on particular and almost accidental convergence of circumstances and exigencies, not indicative of overriding political policies, and even less indicative of any one culture's superiority or inferiority to another. In fact in one sense it simply serves to emphasise the failure on "western civilisation's" part to impose any cogent and beneficial policy designed to encourage and place a value on scientific knowledge. Where it happened as a new development it was as much happenstance as planned.
Universities in Europe were to enjoy a rather mixed relationship with their political masters for much of their first few centuries of existence, and were viewed with as much mistrust as any madrasah which might sow seeds of dissent or sedition in the Muslim equivalent. With the consolidation of power under the Ottomans it can be argued that the authorities had more suceess in suppressing these sources of potential resistance - to the detriment of education - and that this led to intellectual stagnation in many respects within the cultures they administrated. This is simplistic, but broadly accurate. However that is not to imply that in the rest of Europe the opposite pertained on the part of political leaders. Political diversity was the chief protector of the trend towards university-led education, not concerted policy.
I prefer therefore to take the continuity of intellectual pursuit in its broadest sense before attempting to pass judgement on any one political or religious system for disruption in its progress. It can be the beneficiary temporarily of certain systems' protective measures, but its survival has never been down to one such intervention, and its overall progress has been despite the many such systems' rises and declines along the way, not because of them.