Anandtech did a review of the Pentium M (Banias & Dothan) compared to the athlon 64 (1800Mhz) about a year ago.
The Turion is basically a lower power (not performance) version of the athlon64. Performance will be the same as the desktop processor (at the 2800/3000/3200 clock levels).
The pentium-M comes out on top for application performance (like wow, I really need MS word to run faster!).
Athlon comes out faster for Multimedia/encoding/games, but as usual, not by much.
Point here is:
1) AMD has support for 64bit.
2) Turion supports dual channel memory which will give it better performance.
3) Pentium 4 has lower power consumption. (hence longer life or smaller/lighter battery
4) Performance is similar regardless.
1) Pentium4 will have longer battery life or a lighter design.
2) The HD will be the main issue here. Laptop HD's have a platter size 2/3 the size of a 3.5"HD, and based on 2*pi*r, that means that HD performance will be between 1/2 & 2/3 of a desktop based HD, aka 15-25MB/s vs 30-45MB/s(desktop) @ 7200 RPM.
Most laptops these days come with a 5400RPM HD aka 12-20MB/s, which is nasty. 4200 RPM HD's, I dun even want to think about.
3) I have yet to see any laptop with a dual memory interface. They generally only put a single Dimm in the laptop & expansion capability for a single dimm.
My personal choice would be:
1) Turion, as the battery life would be less important to me, most of the time it would be operating on mains power.
2) Turion, as I'm 6"2" and the extra weight of a cheaper laptop wouldn't make much difference to me, over short distances.
3) Turion, cos it can use windowx64 & 64b app's as well as current 32b.
4) Turion cos Intel charges a premium for chips which are negligibly better, hence a turion laptop will likely be cheaper.
5) With the money saved over the Intel chip, I'd upgrade the laptops HD to a 7200RPM (& bigger) model.
But Dizzy is correct. For most people, the Pentium-M will be a better choice.