The only board that'll fit is a 1737 board - nothing else will fit.  It uses the same Core 2 CPU your system does, so there's not much of an upgrade there - yes,  you can fit 1066 MHz bus CPUs to the 1737 vs. the 1735, but the Core 2 is not exactly cutting edge.   Neither is the video chip much of an upgrade.

You're better off leaving the system as is and putting whatever you'd spend on such a modest upgrade toward a newer system - this one's a dead end technology-wise.

Another 1735 board would fit, but wouldn't be an upgrade.

The 1735 and 1737 chassis/boards are interchangeable.

Well any upgrade for cheap is good for me. I dont possess the money for a 600 dollar laptop. I'd rather spend 60-100 dollars upgrading the hard ware on this one. The Intel Core 2's suite my needs just fine. I just don't understand. How wouldn't another 1735 board fit in my 1735 case? Isn't there differences between the 1737 and 1735 layout?

But if I interchanged my base model board for a board that possess persay the ATI Mobility Radeon HD 3650, which is a lot better than the base integrated Intel Express Chip. Which for my laptop would be a big upgrade.

Bump

Your original post made it sound like you wanted to upgrade from the AMD 3650 GPU.  If you have Intel video now, you can upgrade to the AMD (formerly ATI) version - you will need the mainboard, a new heatsink assembly (the original won't fit the AMD board) and if you have a 65W power adapter, you'll need  a new one of at least 90W (65W was commonly shipped with UMA Intel boards;  it won't work with the AMD version - you need at least 90W).

I have a collection of Dell adapters that isn't my concern. But it sounds possible.

Can I ask a question here that follows this thought?  I assume a 1735 motherboard replacement would use the same Windows installation but would a 1737 replacement, or would Windows have to be reinstalled?

The two systems are very similar  -- it may work particularly with Windows 7.  It likely won't if you've upgraded to 8.1 or 10.