Is the "hand-written decree" (I believe you're talking about the minutes of settlement) signed by both parties and endorsed by the judge?
If so, then it's like a rough draft of the court order. The actual 'good copy' is being made to reflect the minutes of settlement word-for-word.
It is my belief that your ex can go against the order if she so chooses. She can take the children to BC without any repercussions... unless you choose to file a notice for contempt motion (form 31:
http://www.ontariocourtforms.on.ca/forms/family/31/FLR_31_Sept105_ODA.pdf) explaining to the judge why and how your ex is in contempt of the court order. If this is the course you wish to take, you will need to contact the courthouse and ask the court clerk to schedule a court date to hear your case. Once you have the date, you fill out the form, serve it on your ex, file it (along with your affidavit of service) with the court, and leave it in the hands of the judge on the day of your court hearing.
If you don't mind me asking... what is the reason behind the judge order that neither parent shall take the child out of the province? And what is your reason for preventing the trip? (You will need to have very good reasons - with proof - as to why your ex should not take the children out of Ontario.)