One important formal difference is that self-represented litigants cannot claim the same costs if they prevail in a civil case. Costs are capped at 2/3 of what they would have received, had they been represented. ("Disbursements", such as court fees, are not subject to the same limit.) Moreover, their deemed hourly rate is £19/hr, unless it's possible to demonstrate that they've incurred greater financial loss as a result of doing the work. This is much less than the billed rate for qualified counsel.
A less practical privilege is wearing gowns and wigs. Self-represented litigants are not allowed to pretend that they are barristers by donning their finest horsehair, or going on about "my learned friend". Solicitor-advocates are "my friend" but laypeople should not use this language at all.
Outside of the courtroom, ordinary people can in principle do their own conveyancing (for example), but there are many practical obstacles. One is being fully exposed to the costs of mistakes. Mortgage lenders will often refuse to let random members of the public take care of the legal intricacies, and some solicitors on the other side will advise their clients not to bother. There are also a few technicalities which are easier for practicing solicitors, such as access to DX (a specialist private postal service) for shuttling reams of paper documents across the country. Case law (Domb v Isoz  1 All ER 942) lets solicitors effect an exchange of contracts by telephone, but this has not been recognised for other people, who continue to have to do it physically.