The benefit of the miniature-specific paints are that they are ready to go right out of the container with the right medium for painting miniatures and a decent amount of pigmentation and opacity.
You can get similar cheaper results by adding a flow medium to lower-end artist-quality acrylics. These aren't the "craft" paints that cost $1.50 and come in the 2oz bottles, these would be brands similar to the Liquitex Basics line of fluid acrylics. The difference is that the artist paints have more pigmentation so the colors are more saturated and give you better coverage.
You can also go one step cheaper and use the craft paints mentioned above (Apple Barrel, Ceremacoat, etc.) with either a flow medium or just distilled water to thin them out, but it's very hard to get deep saturated colors that way, and it can take several coats to cover over dark-colored mistakes with a light color. I personally find it's just not worth it, but it's how I started out and it was good enough to let me know that I liked painting minis.
For a cheap alternative to artist-quality flow mediums, you can try Flood Floetrol, it's sold at most hardware stores, costs roughly $7 a quart. A quart is about 5 times more than you'll probably ever be able use but I haven't seen it in smaller quantities. It's made for latex based paints, but it works fine with acrylics. The benefit over water is that it doesn't dilute your colors nearly as much as water. You can generally get away with 2 parts Floetrol to one part artist paint before you start to lose color, although you'll probably find that you want much less than that for the best ratio.