I sat in a very dull meeting about this before Christmas and this is what I remember. Essentially, the UK are quicker to authorise vaccines because they work with firms early and are willing to pay in advance. The processes are really well worked out, not sure why other countries aren't so organized. Moderna vaccines take longer to manufacture, and the firm has limited supply lines beyond the US, which poses issues because they capped the amount you could pre-order prior to approval meaning states ordered 5m max (UK got 7m, so I assume more money was paid) and then has to bid again for more doses. It caused a lot of frustration for a number of governments who missed a trick with the Pfizer preorder e.g. EU. Ultimately, we don't like big business but they're much more efficient and effective at this sort of thing. It caused a lot of friction between the UK and US/EU who were slower off the mark. The same happened with PPE purchasing when the UK bought so much of it, it meant other governments were unable to get any.
There are also some differences between efficacy, hence the order in which they are delivered is important and to which groups. Pfizer is at 95%; so frontline workers likely to spread, and in significant demand as well as elderly who are most likely to have mortality concerns get the 20 million vaccines we have from them. We have 3.5m vaccines from Moderna, which is 94.1% effective - but slightly less in elderly people. So this is great for later groups. AstraZeneca offers 90%, but requires a long waiting period between doses which means people are at risk between those times - thus this should be used for people able to live independently for some time, working from home etc.
AstraZeneca is the one we are purchasing for impoverished countries due to the easy storage and low cost. It's going to be the one that sells the most. Meanwhile, I had the Sinopharm vaccine a few months ago. That's at 79% efficacy in one trial and 86 in the one I was in, but have since had the Pfizer vaccine too. Problems may come as countries double dip and vaccinate people twice after having first used less effective options on critical personnel resulting in global shortages. There is also a Russian vaccine, Sputnik V, but unlike the Chinese they won't allow independently manufactured trials so no one trusts the 92% effectiveness data.
Also -- while I remember, not all countries have independent approval processes. Many piggyback on decisions made by the US or UK. So when we buy vaccines for countries e.g. Yemen, basically there is no 'approval' process at all.