So your local authority will probably have a regional adoption agency. They are basically the ones who get "first dibs" on placing any child placed for adoption. They tend to have a broader acceptance level of adopters because they will have in house placements.
Voluntary adoption agencies tend to be more selective. If a regional adoption agency can't place within their area (which, remember, can be quite large, so don't be dissuaded), they can essentially pay for a VAA to arrange a placement and support the placement after the child has been placed. That means VAAs are more likely to have more stringent criteria (they don't get paid unless they find the right match and so they need to be sure that all adopters are likely to find a match), but also able to offer more specialist support. They sometimes have specialisms, and more targeted groups they work with. They can place children nationwide through something called link maker, basically a website where you match potential parents to children via their social workers. If you can't find an appropriate match withing a certain period of time with them, you will be given access to linkmaker to look for a match further afield. Usually this is a couple of months after approval, or it is in our area.
It's worth remembering that a child may be considered hard to place for a variety of reasons, including local connections, lack of diversity in the area or a lack of suitable adopters known to the RAA- because the RAA handle adoption training and adoption of children, they have a reasonable sense of who is coming through the system at both ends and will know whether a child is likely to find a placement within their own pool. It doesn't necessarily mean they are intrinsically more difficult. They will use link maker to find matches.
We went with an RAA because they were less concerned with BMI (some VAAs basically said if you were even a bit overweight, you couldn't adopt with them), and we didn't mind a local placement. However post adoption support can be more patchy with RAAs, and obviously on a smaller scale than some of the VAAs can manage.
Look carefully at what support the offer post adoption is, and think about what type of child you would like to adopt. From my experience, RAAs tend to place younger children or those with less complex backgrounds internally, and try to limit linkmaker to more complex cases (siblings, medical needs, transracial placements are some examples).
Everything said above is a very broad generalisation, so please don't take it as gospel. There will always be someone who has had a different experience or views things through a different lens. Also remember that adoption support post placement can come from a variety of other places, so it isn't just limited to the agency you go with. Don't dismiss local support- we have found it invaluable where offers have been local, because travelling with kids can be an absolute nightmare. Knowing where to find a decent OT, or a nurse who knows their stuff about FASD in your local area, or which primary schools deal really well with trauma is not knowledge to be sniffed at.
I can't say I have experience with any of the agencies above personally, but some close adopted friends have had quite positive experiences of Barnardo's.