In all the videos I've seen, the Vampire Lords only use the new spells that they have access to. Does anyone know if they can do traditional magic like Destruction spells and conjuring atronachs, or can they only use vampire spells when in Vampire Lord form?

From the looks of it, no. They only can use Lord powers

From what I can tell you have to be in normal form to use normal magic.

Ok. I guess that makes sense, because there would be no reason to stay in your default form during combat as long as you don't have huge enchantments on your clothes. Werewolves can only do werewolf things in beast form, too.

Todd said you can use normal magic with your vampire powers.

Do you have a source?

EDIT: I do remember a gameplay video where the vampire lord player brought up a shortcut menu to select one of the vampire spells and the list had the regular spells on there too

Well, until we have a source, I'm going to guess the answer is no. It'd be pretty broken if my archmage could simply decide to become an all-powerful vampire lord and use all the spells he already has. Besides, that would only give magic users an advantage--it's not like the vampire lord can go around swinging a battleaxe or mixing potions.