The Lanarkshire club have confirmed the resignation of both Brown and assistant Archie Knox after they reopened talks with Aberdeen chairman Stewart Milne.
It read: "We are surprised and disappointed by the decision of our management team to leave the club at this time and in this way.
"But it is not in our ethos as a club to stand in anyone's way that wanted to be somewhere else.
"We also have far too much respect for Craig and Archie and what they have achieved in the game to do anything other than wish them happiness in what they are setting out to do and to thank them for their efforts in the last year.
"It is very difficult to understand their decision but we must respect it."
The statement went on: "Separately we do wish to make clear that as a Board we believe that the conduct of the board of Aberdeen in this matter has been wholly inappropriate and in clear breach of SPL rules as well as basic courtesy.
"At no point did they inform us or seek our permission to speak to critical employees of our club and to seek to entice them to leave our employment."
The statement added: "We realise the desperation they are feeling and the pressure they are under from their fans for their own performance as a board but to go about their business in this way is a matter of gross discourtesy.
"It is also conduct which is beneath the integrity we would expect of a club of Aberdeen's stature.
"We hope Aberdeen's fans will reflect on that and understand our need to do something about it.
"We have raised this matter with the SPL and will pursue the matter vigorously and by all means."
Motherwell insisted they had done everything possible to keep their management team, who have been in charge for less than year and were working without contracts despite the club announcing they had agreed 18-month deals early this year.
Brown held brief and informal talks with Dons chairman Stewart Milne on Tuesday but Motherwell and their manager himself announced the following day that the approach had been rejected.