I haven't messed with the tuba mouthpieces, but I currently play a Wick 3AL in my B&S euphonium and was able to compare classic and heritage side by side on multiple occasions. For the stint that I played Eb tenor horn in Prairie Brass Band, I was also able to compare classic and heritage when I was choosing a mouthpiece.
The classic is definitely the "classic" big, rich Denis Wick sound. It projects very well and has a rich, powerful sound. For my own uses, I definitely prefer this style on both euphonium and tenor horn. I can see some people thinking the sound is fuzzy or too heavy, especially compared to the Schilke 51D and similar mouthpieces.
The heritage is also a nice mouthpiece, but in my opinion it's a little more agile and nimble than big and powerful. I had an easier time with light articulation on the heritage, but at the expense of raw power. If you play loud on the classic it cuts through the ensemble, while the heritage perches on top. I understand a lot of cornet players really enjoy the heritage line, and I definitely think it fits the British style very well. Maybe this is just me, but it felt like the heritage didn't center as easily.
I honestly think it comes down to personal preference. They're both good mouthpieces, but I made the mistake of buying the heritage without even trying the classic because I thought it looked cool.

If you get the chance to try both, definitely do so. I can also really recommend shopping with Dan from D.F. Music. If you're in the Chicago area, you can make an appointment to head over and try basically whatever you like. He keeps a lot of models in stock, and he's pleasant to deal with.
Slightly off topic, but for posterity's sake I should also mention the Steven Mead models since I've also owned them. Personally, I have to say I prefer both the classic and heritage over these. The original SM models give you this massive sound, but as soon as you start to run out of air it dies almost completely. I had a lot of difficulty playing soft passages and maintaining a good sound quality. The new Ultra line is completely different. Still keeps a lot of the power and adds some agility, but a very bright sound compared to any other DW in the same size range.
PS to nycbone: The confusion over heritage vs. classic in the tuba line is because the "heritage-style cup" was initially designed to replace the old classic funnel design. I guess there was some backlash, so DW rebranded them into two separate lines.