In some ways LA's and VA's are similar, in some ways they're a bit different
The gap between them has lessened somewhat in the last few years as well - it used to be that, because VA's placed children that LA's couldn't find homes for, VA's tended to deal with adoptions of hard to place children. So if you wanted to adopt a young white child without significant additional needs, you were better off going with an LA. However in the last couple of years I've seen adoptive parents adopting these young healthy babies through VA's, so it seems that things are changing and LA's are actually more likely than they were to use VA adopters, especially when the VA is part of a larger consortium of agencies within that area.
If you approach a VA, they should tell you whether they think it will be possible for you to adopt a child aged x with y needs, through their agency.
As has been said, the process until approval should be mostly the same, with the exception of one or two agencies - I know for instance that the VA i-adopt have done it a bit differently since they started. However it's the same principles of background checks, training, assessment and approval panel everywhere, however they choose to do it.
VA's can tend to be more efficient and have less delays in the process, because they obviously have more resources and time to dedicate to approval.
Post adoption, agencies vary in their support. Traditionally, VA's were often a bit better at providing post adoption support, again they had more money, time and resources. However, I have heard some adoptive parents say bad things about their VA's post adoption support as well as LA adopters. VA's vary in the quality of support they provide, just as LA's do
If the person you know is having difficulties with a certain social worker, it may be an individual thing ie. it's just this SW, not an agency problem. However, some agencies are more welcoming than other! Although I personally adopted through my LA and only ever talked to them, I would recommend other adopters talk to multiple agencies so they can go with the agency which is the best one for them.
It does depend where in the process this person is. If it's right at the beginning (ie. initial home visit, getting information stage), she/he/they can cut their losses and go somewhere else easily. However once the homestudy starts, you can't just go somewhere else - the second agency will ask the first agency why you quit the homestudy etc, and you may have to answer difficult questions.