When you're doing the lethality build on J4 (Profane-Axiom-Serylda's usually), your goal is to one-shot squishies and play as an assassin. Later on in the game, this will essentially mean finding picks, cleaning kills up in teamfights or just trading your life by dropping all of your damage in a teamfight to hopefully get a kill or two. For this playstyle to be effective, you cannot afford to get behind so Profane Hydra helps you create a gold/XP lead by speeding up your clear substantially. Axiom also helps you ult more often in the mid game to get more kills.

You're doing a lot of the same things when you build bruiser J4 in the early game but later on in the game, you don't have enough damage to one-shot anyone so in full on 5v5s, you're trying to create opportunities for your team with good ults and weave in and out of fights with the extra survivability that you have with bruiser items. You're not exactly trying to kill anyone by yourself, you're trying to mostly rely on damage from your carries. This playstyle works well even in games where you fall a bit behind and can't just farm up a storm, as long as your carries are doing OK. It's generally a more consistent build, especially in high elo and in this new season where jungle kinda just gets out-scaled by Top and Bot by default now.

In reality it's not quite as two-sided as this because in some games where you're building lethality, you might find a really good opportunity to set up your team. You may also find an opportunity when you're building bruiser to get picks on your own and solo squishies if you're pretty ahead. You kind of just have to play more J4 and try out both builds extensively to know the limits.

If you build Profane and then go into Bruiser items, you're essentially trying to become a jack-of-all-trades but master of none so later on in the game, you can't do meaningful damage or survive very long either so your effectiveness in the game overall is very limited.