Is a digital cable converter box and digital TV converter box the same thing?

No, these are different things, although they are similar. Both take digital signals as input and convert them into video and audio signals your TV can output. But the type of digital signal coming in is different. In a cable box, the digital input signal comes from the cable; it is called QAM format and it might be scrambled (the cable box can unscramble it). In a TV converter box, the digital input comes over the air from an antenna, it is called ATSC format and it's not scrambled. So if you tried to use a TV converter cable box for digital cable, you won't get any cable channels. That's because the box is expecting to see ATSC signals from an antenna, instead of QAM signals from a cable. However, you still might get some regular over-the-air channels, because sometimes the cable will act like an antenna. Most new TVs have the ability to accept both cable (QAM) and over-the-air (ATSC) digtal inputs, but the only type of QAM they work with is called "clear QAM", which is unscrambled. Therefore since cable companies typically scramble their channels, you probably will still need a cable box, since the TV can't unscramble the cable channels.