What makes tape sound better than vinyl ?

In regards to a cassette deck, what would be your recommendations based on current good used equipment still in circulation?
Might be a fun worthwhile endeavour especially as I still have a huge pile of tapes from when I used to play them in my muscle cars before switching over to Hidden Stereo systems which were then cd/ fm based not cassette/fm based as the old indash spindle fit systems were.
Not sure on budget as have not considered tape in a long while but my guess is that $300 to $350 should get something halfway decent?

Thank you
Kevin

Kevin, I took a look at what's available on ebay right now. I can see that prices for good decks are going up. Of what I saw the closest to your price range that I would bother with is Nakamichi 582 for around $600 including shipping. The seller promises to service the deck before shipping it. Those old decks really need service form time to time. I have no personal experience with Nak 582 but it's good. I read a lot about cassette decks years ago before choosing Nak 682ZX, which is I believe the best performance value. Couple of extra hundreds of dollars go a long way when it comes to cassette decks. But if you are very serious there is a rare Nak ZX-9 for $1400 serviced by Willy Hermann. That's where I service my deck. He currently does only overhauls which is in fact not full overhaul but partial overhaul, including all the calibrations, alignment, transport disassembly etc. For my deck he charges about $650, I do it every five or six years. 

@inna 
Many thanks for the info,I would have guessed Nak would likely be the way to go but it seems my price guesstimate was out a few dollars!
Have to be fairly serious in my head before spending near 1k. Especially as I just bought the Nottingham a few days ago...lol.

Kevin

I am thinking for my budget and intended very occasional use that a lower model may be in order. Like the cr3a,4a or 5a.
At present I am thinking of using mostly to play existing collection of tapes( I also have a friend who has a vast collection of tapes and no deck that I am sure I could "borrow" some), rather than record anew.
Partly due to my DAC/pre has no tape out function so would have to go direct from phono stage or CD player or vault to the deck to record anyway.
Decks along these lines seem to be $250 to $500.
Lots to consider before jumping in!

I am not familiar with them but CR4A and CR5A should be okay for playback. I think, they were made in Taiwan not Japan, and they will not have classic warm Nakamichi sound.
Tandberg 440A is good, should be around $500, I am not sure about its reliability, though.
Recording directly from phono or cd player is a way to go. Good interconnects make a difference.