Apply for the more challenging position. Don't shy away from the pressure or scare yourself. Try and fail but don't sell yourself short. You are an associate apply for that position, let them decide if you are qualified, and if they decide not ask to be considered for the lesser position. 

I've seen firms calling people "design associates" because they weren't licensed.  They're not associates in the firm management/ownership sense.  I guess the firms just didn't want to call them interns or designers.  

Are you sure this isn't the case? 

Large firms typically have corporate titles (Associate, Senior Associate, Assoc. Principal, etc.) that are different from people's roles (Architect IV, Interior Designer I, etc.) where there are different sets of qualifications for each.  Corporate titles are usually negotiable to a point, but your role is directly tied to your job function, skills, and experience.

TBH, if I came across two job listings for the same company - one that said Associate and one that said Architect - and the requirements were pretty similar, I'd assume it's the same position with no other info available. They probably just don't know what they're doing or there is some internal miscommunication or lack of communication.

They mean associate as design "associate", not becoming owner of the practice....are you sure you know what youre doing? The architect role would be someone licensed, the associate role would be inferior to the 'architect' position.

As you can see from the responses, and from your own post, "Associate" is not a standardly-defined title in architecture firms, and can mean many different things.  In some firms "Associate" is what they use to describe everybody who isn't licensed, including part-time student employees and people fresh out of college, while in other firms "Associate" recognizes employees assumed to be on the partner track - which may or may not have anything to do with a project management role or a particular number of years of experience or tenure with that firm.  The AIA has recently complicated this, by endorsing "Architectural Associate" as the new replacement for what was previously called an "Intern" - even though "Architectural Associate" is an illegal term for an unlicensed person in 40+ states, due to the inclusion of a word derived from architect.  So Associate is a pretty meaningless distinction without familiarity with the inner workings of that particular firm. 

Apply for the position by writing a cover letter that ticks off your qualifications, framed in a way that aligns them with the job responsibilities listed in both of their ads.  Call yourself an architect with 10 years of experience managing project teams.  Don't be specific about which ad you're responding to.  Once you get an interview, discuss with them what they're looking for, and find out more about the firm.  Negotiate position and title from there.

We have 5 people with under 4 years of experience, all working on their IDP (what's that called now?)  There's an argument in the office about whether the term intern should stick.  Then we have a gap, and the rest of the 100 person firm has at least 9 years of experience.  They're all at least associates, but not all architects.  If you get hired young here, it's expected that you become registered before you garner the associate title, and at least in my opinion, if you get registered and don't get the title, it's time to leave.  

I've been arguing that we're really top heavy, and anyone with a project manager role should be able to handle redlines/instructions for at least 2 interns... but the firm thinks we need to hire more project manager level architects, not interns.  We've at least gotten to the point that you won't be hired if you can't revit your own permit sets as a PM.

Associate is a rank within a firm's organization.  

Architect is a legal, professional designation granted by the state.  And, it seems, it can also be an organizational rank in some larger firms.

I suggest entering a big practice at the absolute highest position they are willing to give you.

Don't plan on coming in at a low level and then being promoted.  Big firm management doesn't always fairly recognize talent and contributions to projects.

Apply for position where you can, perhaps with some difficulty, fulfill the duties listed.  One school of thought is 'fake it til you make it', though I personally find it dangerous and irresponsible to claim you know how to do them, you can be more candid in the interview. Most firms realize the shopping lists are unrealistic and will look for candidates that check off most boxes and show promise.

Titles are meaningless other than to specific firm's hierarchy, and even then, they often are simply a way to keep employees happy.