So what is the real difference between the jobs? Just the title?

It seems like, if both companies measure seniority the same way, then you're qualified for one of the roles, and under/overqualifier for the other.

Or are you really a senior developer by level, but applying for an intermeiate role so you can go back up to serior and feel like you're growing?

Imho, there is no difference in "growth" between a senior and an intermediate developer role. Skill-wise, the difference I see is in the type of skill you can grow in. For an intermediate dev, that skill set is probably improving their architecture and pattern knowledge, writing better code and making it more stable by learning new concepts. For a senior developer, the skill set shifts more to inter-personal skills, like how to guide an individual or a team, how to mentor someone, how to work out loud, how to intermediate, etc.

If you mean growing your career, I also don't think that a senior dev cannot grow further. There is always room for becoming a manager, or changing position and doing something else (for example go to UX); or changing the company, going to a bigger one, which might pay better or one which needs a slightly different skill-set. Even the PotUS has room to grow (after all he is not the PotW :D)

However, if I had to decide, I would prefer a senior role, with more responsibilities and more freedom in the bigger picture, over an intermediate, who can tinker and play around with the details.

Do you mean personal growth like skills development, or professional growth like there's nowhere you can be promoted to from that position?

I'd love a role that didn't have a path-to-promotion as long as it let me build and work on skills I already had!

It is very difficult to answer this, but I would go for a role that makes me happy and gives me sense of learning while also providing value.