"Don't update your profile"
This requested delay, in my opinion, matters little. Your director likely does not want clients to see a current team member leaving the company, and ask questions ("why would somebody leave the company I'm paying? Is there a problem?"). A possible "fear" is that those clients to which you're connected would ask why you left, but the director could probably manage those questions. A much more dire fear is that clients will contact you, and this could be bad for business, depending on your answers.
I say it matters little because a few weeks could allow for a new hire (replacing you), which would at least appease clients, but if the business has problems that would lose clients based on a single employee leaving, your director likely has much bigger issues that won't be fixed in just a few weeks.
Challenging that your new boss may request that you update your profile was a smart move. Unfortunately, it triggered a strange response:
"Disconnect from our clients"
I assume that you're moving to a related company in the same field; essentially, a potential competitor. If a client sees this, they could ask you questions about why you switched, or discover this new competitor if they'd never heard of it. Questions here cannot benefit your current employer, so your director wants to avoid any interaction between you and your company's clients that could hurt business.
Solution: do not comply.
- Don't wait to update your profile. You've already posited that your new boss could require it, so it's unlikely that your director will hold this against you.
- Don't disconnect from clients, especially the ones you're friendly with. Your director seems not to have thought this through, since disconnecting will likely worry clients even more than merely seeing that you left the company. In any case, maintaining connections with clients is important. The impact of having friendly connections with what would be a competitor's clients makes you a very valuable employee.
Don't worry about your director somehow attacking you or your reputation because you didn't do as he asked. This would be immature at best, but is easily maneuverable: if a potential employer sees a bad review by your current director, you have good relationships with clients to prove it wrong! This would reflect poorly only on your director.
You are leaving the company, after all; assuming you do so cordially, you need not comply with your director's requests, and there is very little chance of any (successful) retaliation.