Both Pfizer and AstraZeneca are delivering the vaccine to the EU more slowly than they promised. In Pfizer's case because they need to partially shutdown the factory for a couple of months to retool, and in AZ's case because the factories they are opening in the EU are running behind schedule. Normally that would just be annoying - these things happen, and everybody is doing their best.
The EU, perhaps rightly, didn't place such a high priority on vaccines as their near neighbours (us). That meant that they were always going to be a couple of months behind us from the get-go. If we weren't nearby providing a yard-stick of what a fast rollout looks like the current EU rate of vaccination would be fine, even though the virus has flared up this winter more than the EU countries were hoping or expecting, and is causing them massive ongoing political tension. The fact that we are there though is making EU leaders, and the EU Commission in particular, look bad. In that circumstance, it looks particularly bad that we are getting more supplies right now than the entire EU is getting, and will be vaccinating some of our distinctly less vulnerable people while they still have unvaccinated over-80s.
The response is a collection of howling anti-British negativity, leading to claims that the "British" vaccine isn't all that good, which is just easy-to-ignore nonsense. At the same time though, they are trying to hoard the Pfizer vaccine (which isn't terribly outrageous, but isn't particularly helpful from an organisation dedicated to promoting trade and international cooperation). Alongside that, they are making demands that Astra-Zeneca divert some of the supplies they are sending to the UK so that they are failing to deliver on both the EU and the UK contracts roughly equally. There is probably an argument for that, particularly once we have vaccinated groups 1-4, but it's a far more difficult argument to make when they are blocking Pfizer from supplying both the UK and Canada.
Unfortunately the EU has a particularly mediocre Head of Commission at the minute, and a pretty bad Health Commissioner, and between them they are running around like headless chickens making everything worse.