I have an iPad1 and use the connector from that to HDMI to HD TV and it works really well. I have Apple TV now (£100 vs £25) but still use the cable from time to time, eg I take it on holiday/business trips so I can watch movies held on the iPad (of iPhone) on the hotel tv also if going to friends house and want to bring a movie.
Apple TV picture isn’t noticeably different for me vs the cable but the ATV offers more, obviously it’s wireless, has direct web access (eg redbull tv, ntflix etc). Also the ATV can do full screen mirroring of iPhone/iPad or be another screen for the computer (Mac). For example I click “watch later” on all the Vimoeo midweek mini movies then watch them on big tv via ATV Vimeo app when I’m home.
If you are thinking of an ATV I would hang on a bit as there are rumours of a big upgrade before year-end (as always with Apple)