The Queen certainly does have power, including the power to sign international treaties and deploy British troops abroad. It's true that most of these "royal prerogative powers" are today exercised by government, but that in itself is a serious problem. These powers have been transferred directly from the monarch to the prime minister and don't need the approval of parliament, effectively shutting out the British people from important decisions. That is fundamentally anti-democratic – and it can only happen because we have a monarchy.
The Queen and Prince Charles also have the power to veto bills that affect their private interests. Official legal advice makes clear that Queen's and Prince's Consent (as the "royal veto" is officially known) is not a mere formality. The process by which consent is obtained provides a clear opportunity for the Queen and the Prince of Wales to influence the shape and content of a bill before it reaches Parliament.
Then there's the problem of parliamentary sovereignty. At one point all the power in the land was held by the king or queen. Over time that power moved to parliament and is now held collectively by 650 MPs. However, the fundamental nature of that power hasn't changed – parliament can make or scrap any law it likes, just as the monarch could in the past. This means our freedoms are never really guaranteed because parliament can always decide to remove them. Again, this a direct result of having a monarchy.