A food processor is extremely useful and makes certain large projects easy, which otherwise might be nearly impossible. It's an invaluable tool for grating, chopping and slicing for parties where hand work would be very tedious. They can be used to make good pastry dough, and many bakers better than I use them routinely for regular sized projects. Personally, I prefer to make pastry dough by hand.
A processor uses a wide blade and a wide (large diameter bowl).
A blender doesn't use a bowl at all, but instead a pitcher. The blades are comparatively narrow. Blenders handle liquids and semi-liquids better than processors. They are invaluable for preparing all sorts of drinks and sauces.
Most people, I think, would choose a blender over a processor if they could only have one. However, if you've gotten by without one so far, you might be interested in starting with a "stick blender" or "immersion blender" which is another useful tool for handling liquids.
My recommendation is, if you do purchase a blender, that you bypass the "residential" models and get a professional quality "bar blender" for a little less than $100 -- either a Hamilton Beach 1/2HP "wave action" or a Waring 3/4HP. The difference in terms of what you can and can't do with a pro versus a home version huge. Smoothies, milkshakes, purees, you name it, are all specialty quality.
On the other hand, you don't have to go to the "pro" extreme for a food processor. The best are KitchenAid and the larger Cuisinarts -- with a big edge to the Cuisinarts. If you're counting pennies, you don't need a lot of features. Stop, start, pulse. Add a continuous feed top, the basic set of discs, and you're good to go. It's been a few years since I researched processors, but the last time I did, the lesser machines were quite a bit lesser. My feeling is that a Cuisinart is the way to go.