Colors don't show up well with a projector when lights are on--contrast and color depth are typically lost.

You will also need several feet of space between the projector's lens and the screen--typically no less than 5ft, though 8-12ft is better, especially for the size you are looking for (keystoning can only do so much).

You can easily get an Epson PowerLite projector for $650-$800, but keep in mind that a ceiling mount will probably run about $100-$200 on top of that, and then whatever a screen of the appropriate size costs ($75-$250). Putting a projector on a table will save on cost, but you will likely have to re-adjust it frequently as you set it up or take it down, or as it gets moved or nudged. I wouldn't recommend this. If you go with a projector, have it mounted.

Also, here is a throw distance calculator to help you with positioning of Epson projectors: https://files.support.epson.com/Epson_Handbook/html/interactive_distance_calculator.html

A 60" TV in the sub-$800 price range will be a bit harder to come by. The only brands I can recommend are Vizio and LG (though you might have more luck finding a Vizio in your price range). Don't bother with Samsung--the power boards in those TVs tend to die after a few years. I can't really comment on any other brands. Note that if a TV is an off-brand that is *too* cheap, you run the risk of it dying early and having to buy another one, so you wouldn't actually save anything in the long run.

Overall, you can get a larger size screen with a projector for less money, but you typically sacrifice color contrast/depth/brightness. A TV offers better color contrast/depth/brightness, but tends to be more expensive the larger it is.