While they look alike, they are very different and it is immediately apparent upon using them.
As others have mentioned, the shutter speed on the Olympus is set with a ring around the lens mount. The pentax with the knob on the top.
The Olympus uses a cloth horizontal shutter, the pentax a metal vertical shutter.
The Olympus was made as a pro camera and the mx as a mid-level.
The Olympus used a mercury battery as stated above.
The Pentax MX uses a horizontal cloth shutter.
Pentax did market the MX as "The Little Professional", though it was not promoted that way as intensely as the OM-1 was. The OM-1 had a much larger "system", too.
As stated in another post, the MX does not have mirror lockup. But the mirror can be "pre-released" on most of them by quickly flicking the shutter button with a finger.
The MX has shutter release lock-the OM-1 doesn't. The meter in the OM-1 is the basic CDS cell, center the needle type, while the MX has gallium arsenide (similar in function to silicon cells) with an amplifier circuit outputting to LED's.
In the OM-1 the galvanometer for the needle is inside the ASA knob on top, a very clever way to reduce size while retaining a standard form.
The physical similarities are superficial, and if you had them in front of you, you could easily see (and feel) how different they are. The MX is a little smaller, with a lower top deck. That makes it hard for some people to grasp it comfortably. They are similar in specification, with the same shutter type, shutter speed range and X-sync speed, both are designed to be light and compact, both take winder or motor drive, have a huge, bright finder, etc.
They are both nice machines. As a Pentax user, I have a couple of MX's, but I think the OM-1 handles better. MX shutter knobs have heavy click stops and are too short to grasp easily. I find the OM-1 layout easy and quick to use. On the other hand, the MX's shutter speed and aperture readout are nice to have at times.