It depends on other factors really. The solicitor needs some kind of "proof" that the person approving electronically is the right person, i.e. guard against an imposter using a fake email address etc., so that comes on the back of the due diligence and identity checks and whether there's been face to face contact, previous mailing of documents in the post with a "wet" signature (i.e. letter of engagement, pre contract documents, etc), proof of the bank account/funds transferred (i.e. evidence of account holder and bank account), etc etc.
It's all part of a big jigsaw puzzle of evidence and due diligence. If the solicitor is happy in all other ways re identity checks, money laundering, due diligence, etc then they're going to be more content with an electronic approval. However, if the due diligence etc has a few gaps, then they may want a "wet" signature for additional evidence.
It's all about risk - if they're happy they're dealing with the "right" person, and that everything else stacks up, then they'll take the risk with an electronic approval. If not, then they won't take the risk as they won't want a compensation claim against them. Re the point with big versus small companies, again it comes back to their due diligence, risk assessments etc.