To be an SLR the camera has to have a mirror and optical viewfinder as cabbey states. The optical viewfinder has several advantages, namely a crisper image for easier manual focus (as you're getting the full resolving power of the lens, instead of a small LCD screen) and zero lag (as what you're seeing is happening in real time!) which can be useful for sports etc. There are a few disadvantages as well, you can't zoom an optical viewfinder, nor can you preview image settings (you can set an electronic viewfinder to show you the image in black and white for example).

Whilst the viewfinder is the only difference by definition there is another important distinction which impacts shooting, and that is that digital SLRs almost always have much larger sensors than compact bridge cameras like the FZ35. A larger sensor allows you to capture much more light giving the SLR much better performance in low light.

To visualise just how much bigger the sensor is in an SLR here's a visual comparison between the FZ35 (which is typical of bridge and compact cameras) and the Canon 1100D, a typical entry level digital SLR:

Some compacts do have SLR sized sensors (eg. Sigma DP2, Fuji x100), but they tend to have a very limited zoom range. You certainly wouldn't be able to get a 18x zoom on anything but a small sensor! The ability to produce small light zoom lenses is the reason manufacturers have stuck to small sensors.

In addition to capturing more light, the larger sensor also gives you a shallower depth of field (when matching the field of view of a small sensor camera). This means the range of distances from the camera that are in focus is much smaller. This can be used to artistic effect allowing you to isolate the subject and blur out the background.

The only way to get blurred backgrounds with a small sensor compact is to use a long focal length (zoom in a lot), or get really close to a small subject (using the macro or close up function).

I would say shallow depth of field is the biggest difference in terms of the images, as nothing can be done to replicate the look of an SLR when producing images with shallow DoF in certain circumstances (except by faking it in Photoshop which is difficult and time consuming). When I see an image with shallow depth of field, especially if it's a moderately wide angle I can tell instantly it's from an SLR not a bridge camera.