Lightroom gives you a lot more control over the processing of your images. You can finely tune vignetting (add and remove), curves, sharpening, split-toning, adding clarity, removing chromatic aberrations, powerful noise reduction, de-warping (lens distortions and perspective) "selective editing" (e.g. change the saturation/luminance of one color only, or of one area).
As opposed to Picasa which only does desaturation, it's really powerful when it comes to black and white, you can finely control each color separately, as if you were using color filters on your camera (as was often done when using b&w film).
Lightroom is also extremely efficient a recovering details in the shadows and in the highlights. From a RAW file, you can go from a totally white or black picture, to something decent. Recovery is something where Picasa falls short. And when you have a really nicely composed shot, it's nice to be able to fix some lighting mistakes.
Lightroom has a lot more features when it comes to printing, making slideshows and galleries, and most importantly organizing large sets of pictures (tags, metadata, flags, stars). It is extremely convenient when selecting/deleting shots out of several hundred pictures after an event/photoshoot.
When it comes to editing, you also have a lot more options to check your work: dual views to compare images (while selecting the bests), and before/after views to check while editing. You can also have several edited version of the same picture (I often have a b&w version and a color).
The most useful feature to me in Lightroom that Picasa lacks (afaik), is settings synchronisation. I can work 5 minutes on one picture, and copy/paste (sync) the parameters of a picture to 150 others. So I can edit 150 pictures all at once, sync white balance across all the shots in the same lighting environment. And, when you start seeing a pattern in how you setup similar parameters for most shots you make, you can decide to create a reusable preset of parameters, which you can then apply to all pictures when you import them, and then tweak each selected pictures later for finer details (or special case where the parameter might not match this specific picture). If your workflow in Picasa seems slow, it's time to switch to Lightroom.
Lightroom records the history of all changes to each picture, so you can come back to them later, and revert them easily.
You can definitely setup Lightroom to "watch" folders and add pictures to your catalog as you copy them to your Pictures folder. For deletion, I prefer to select and tag all the keepers, and filter out the rest in Lightroom, which has an option to completely delete them (both from the catalog and the hard disk). I shoot RAW+JPG so this comes handy as it deletes both files at the same time (and they only appear once in the Lightroom catalog).
A few years ago, when I started photography, I loved Picasa. And I still do. I recommend it to anyone starting out photography. When you reach the limitations of post-processing of Picasa (mainly in terms of fine control, recovery, parameter synching, and b&w in my case), then give Lightroom a chance -Aperture has similar advantages I'm sure.
Lightroom will take more time to master, but it's definitely worth it when you're an amateur photographer who wants more control and has a huge collections of images (since I got my first DSLR, I've taken more than 100k shots, and I've got close to a TB of images, which I'm appreciating more and more as I learn to master the fine art of post-processing (analogous by all means to printing in a dark room, with an enlarger).
To summarize, if Picasa works for you right now, don't switch.