I have a Compaq Proliant ML350 web server running Debian out of my house. It's been online for a few years. My server is old. Specs are Dual Pentium IIIs running at 1.0 GHz with 1.5Gb of RAM.

I'm trying to get a scope for how that system compares to a more-modern one (i.e. a somewhat aging laptop with an Intel Core 2 Duo with 4Gb memory).

Reason I'm asking is I'm wondering if I should use a laptop to run a web server compared to this beast of an old metal machine?

Please don't reply back saying I should get HostGator or some other web hosting service. That's not up for debate. I have and will continue to run my own server. I'm just trying to get a perspective on the performance differences between the old server and my Dell laptop.

The Core2Duo will by far outclass your old server when it comes to performance and efficiency (read: you will save some money on your electricity bill).

What about longevity? Can a laptop stay running literally 24/7/365 ? I can easily make adjustments in order for it to; but how long could it last without breaking down?

Due to its limited cooling system, especially smaller channels for airflow, you will have to clean out the cooling system more often than with a good ventilated desktop case, but other than that I don't see why it should have limited longevity. If resource usage of your workload does not significantly rise compared to the old system you likely can even run it in power-save mode all the time, letting it run cooler/with less power-consumption.
Though you may possibly want to make some adjustements how powersaving for the harddisk is handled (laptop harddisks tend to spin down often, reducing the lifetime of the disks.