The simple, singular, difference is that architects are trained to look at the entire project holistically [including communicating with the public or constituency, the municipality, GCs, structural and mechanical engineers, landscape and civil, graphics, etc.].  Meaning they are true generalists.

While interior architects primarily focus on (duh) interiors.  They more often focus on communicating with tenants, lighting consultants, material suppliers and fabricators, sub-contractors, etc.  Their role is more specific than general.  

Which doesn't mean one is of more intrinsic value than the other, they just entail and employ a different [and sometimes overlapping] skillset and network.   

Personally, my opinion is that interior architects get to go into more detail sooner in the process, but architects have a greater understanding of a given project holistically.  It's not whether one is 'better' than the other, it's merely about where your interests lay.

Architects design....interior designers go shopping.

Taliesin - you must not know a lot of interior designers.  I have yet to go "shopping" but I have done a good amount of designing.

M.Arch grad here.  Please forgive me for coming off harsh, but this is my take:

The difference?

-5-7.5 years of exclusive and very demanding education.

-3500 required and approved hours of working under a licensed architect.

-7 (soon to be 5) grueling exams with low pass rates.

-Finally after all of that, we get a stamp.....some sleep...  and can legally practice Architecture.

An Architect is legally responsible for the safety, health, and welfare of the public.  We have to pass exams to be able to practice, similar to the way that Attorneys, Accountants, Engineers, and Doctors must do internship hours and pass exams.  A paralegal is not an Attorney, a tax-specialist is not an Accountant, and a Physician's assistant is not a Medical Doctor.  As far as I am concerned, an interior "architect," is extremely similar if not the same as an interior designer, or interior decorator.  Sure there might be an exam, but that is ABSOLUTELY nothing like the ARE's.

In addition to navigating building codes, overlays, zoning, ability to size beams, columns, air ducts, and detailing curtain-walls, window sills, roof penetrations, etc, as well as any sort of penetration through a fire rated wall, and also having to deal with the ever-so-much-fun-to-deal-with-fire-marshal, and people at the P&Z department... In addition we decide the shape of some interior spaces by moving our non-load-bearing walls (which in the end affects our life-safety sheet, and our exit floor plan which requires actual licensed Architect to approve, stamp, and in doing so risk his own livelihood).  We also sometimes pick paint colors, carpets, Furniture, Fixtures, and Equipment for our buildings.  (The list above is no way at all exhaustive.)

The ability for someone who did not go through all of the grueling requirements listed above, nor have the responsibilities to the general public, to place Architect after their name is a disservice, and discrediting to Architects.  

The difference between what an interior architect is allowed by law to do, vs what an interior designer is allowed by law to do, vs what an interior decorator is allowed by law to do... is pretty much.... the same.  It all falls in the:  FF&E, moving interior walls (which Architects have to in the end approve if they change the life-safety sheets), and picking out finishes.  

The markup on interiors work is freakin awesome, don't get me wrong.  Y'all make crazy good money, which is very enticing.  But in the end what interior "architects," interior designers, and interior decorators do is pretty much the same.  They practice a very small subset of Architecture, which is a profession that comes with tons and tons and tons of liability.  Similarly to a Doctor, Accountant, Engineer, or Attorney.

There are a lot of similarities on the approach with Architecture & Interior Architecture, the difference of course lies in the "accountability".  The advantage of an Architect is the knowledge of Structure while the advantage of an Interior Architect is their sensitivity to haptic experience.

I would not struggle in this decision actually.  I would easily pursue Architecture then be an Architect.  This still gives me the flexibility to take on Interior Architecture projects.