Having stayed at both more than a few times, my opinions are below:

The Holiday Inn over the road is better in some respects;

• It looks fresher from the outside
• Some of the rooms are nicer / more modern
• The lifts generally work and are quick (unlike the Doubletree)
• The tram noise can be greater than the Doubletree if your room faces the road.

With the Doubletree:

• I can select my room at the Doubletree in advance.
• The Doubletree has a better bar area and breakfast restaurant
• The breakfast is free for me as a Diamond at the Doubletree
• The reception staff tend to be very good indeed.
• The lifts are slow and temperamental.

However, to aster , I would recommend you just call the Doubletree and check.
They'll have their own interpretation on Ts & Cs that Hilton corporate won't be able to advise on and in my experience this hotel tends to lean towards supporting the customer rather than strictly applying corporate policies.