I've gone through two labors naturally.
For me, contractions feel the same as a strong BH until (and I know this is infuriating)...until they don't anymore. Labor will often start out with contractions that feel very much like you've felt before...they slowly amp up until they're involving your whole uterus, and (unlike BH) starting to pull the cervix up over the baby's head.
For me, the only thing that would strongly indicate those EARLY contractions are labor (and not BH) is the timing of them. Once they start to get into a pattern (every 12 minutes, or every 10, or every 7), then you can start thinking "huh....this might be it."
Of course, the pattern can start up, then shut down. This has happened to me both times, usually in the evenings before bed and in the week before labor really kicks in. What they say about water/rest/sleep/bath is true...if you can get them to go away, they're not labor contractions.
If you do get patterned contractions, take it easy and try not to get yourself too amped up. Enjoy the chance to practice with these low-intensity contractions.
Once contractions really start working (the cervix being pulled back and over the dome of the head), you'll feel it happening. For me, it felt like pressure. Definitely uncomfortable, but not freak-me-out-painful at all. I think keeping certain mantras and visuals (especially bio-based ones
...the actual musculature of the process) helped me avoid fear, which helped me avoid a lot of pain. Contractions for me felt like a wave of pressure. It'll take your breath away the peak, almost like getting run over by an imaginary steam roller or something...but it's so comforting to know that after the peak, it fades and you can rest. So in that way, it's like waves. Really regular, predictable waves.
At transition it gets more intense, especially if you have a fast labor. The breaks can be very short, and your body starts to react to the incredible changes happening in your cervix and uterus. Last time I got the shakes...my whole body felt like it was experiencing an earthquake, starting at my uterus and going down my legs. The contractions were powerful, but if you get there without pain meds you'll have the coping tools to deal with the contractions--you KNOW you can do this for the rest of the time, because you've done so much already. Still, the steam roller/wave feeling can be overwhelming. You just kind of hold on (mentally) and let go (physically) at the same time, knowing and trusting your body, knowing and trusting that this is part of the process of pulling the cervix up and over the baby's head. Once you get to the extreme of it, you're SO CLOSE. It's a very powerful motivator.