Welcome to the Forum Wes,
Sounds like you purchased an OEM DVD.
The OEM product has either the 32 bit or 64 bit DVD not both.
When you order an OEM you need to specify the "bitness".
If this is an OEM product you cannot install it on your new computer per the license.
A full retail product comes with 2 DVD's, a 32 bit and a 64 bit DVD.
You get a license good for either version but not for both.
Once you activate one version you can't activate the other.
You would need to purchase an additional license to use the other version.
If you don't have an OEM disk then you should have recieved both DVD's. because
the Full Retail and Upgrade packages contain both versions.
At any rate the license for either the Upgrade or Full Retail product is only good for one or the other of the two DVD's.
What you can do is purchase a 64 bit OEM disk for your new computer.
Newegg.com - Microsoft Windows 7 Home Premium 64-bit 1-Pack for System Builders - Operating Systems
As you can see above OEM Win 7 64 bit Home Premium can be had from NewEgg for $105.00 US.
Do be aware that OEM versions cannot be used on another computer once activated, as a matter of fact changing the motherboard of a computer with OEM Win 7 installed and activated is enough to trigger deactivation.
Changing other hardware shouldn't trigger deactivation.
There is no support from MS for OEM Win 7 as the original equipment manufacturer, you, is responsible for support again per license.
These restrictions are why the steep discount of OEM over Full Retail.
Hope this helps.
If you install 64 bit Win 7 on your new machine at least 4 GB of ram is preferred and depending on you MOBO and BIOS more can possibly
be added and suported by Win 7 64 bit.
As to your question about any disadvantage to using a 32 bit OS on a 64 bit machine the obvious disadvantage is not being able to take full advantage of more than 4 GB of ram.
A 32 bit OS cannot address more than 4 GB of ram, it's not possible mathematically.
Avoid an inplace upgrade if possible. When the time comes backup you data and format and perform a clean install.