The biggest advantage of the likes of BA and LHT is they can offer a complete package, a "one stop shop" if you like for maintenance services for an aircraft operator.
That is, they can offer line maintenance, heavy maintenance, spares, engineering support, CAMO services etc.
This of course can cost a fair bit more than some smaller companies.
Their biggest disadvantage (as with most large companies) is probably the lack of flexibility. For example, you are flying to a new airport and you want them to set up a line station, they might not be too interested if it's only for a couple of aircraft per week, a smaller company might be more inclined to jump though the necessary regulatory hoops to set something up for you.
Another example I've seen several times is the lead time in the case of an AOG, unless you're lucky enough to have your aircraft AOG at one of their main bases it can be a problem for them to get the required personnel to a remote base, in fact I did a job yesterday for LHT for exactly this reason. Again, the smaller maintenance providers can often arrange something at much shorter notice.
With the fleet of aircraft expanding worldwide there will be an increased need for all aspects of aircraft maintenance, dependant upon the needs of your operation will dictate if the likes of BA or LHT will suit you or possibly you might be better off with one of the smaller maintenance providers.