Rear Diffs don't smoke like that. There's not much oil in them to burn. Neither do transmissions really. If that's the only symptom, I'd say it's unrelated. Maybe a shop guy had oil on his hands and bumped your exhaust while he was down there... 150 miles is a lot for an incorrect tire mismatch. Subaru only allows the spare to be used for a maximum of 50 miles (but again, the spare is a more extreme mismatch). It wasn't good for it, no.
The first number is a width, in millimeters.
The last number is the wheel diameter, in inches.
Most people are OK understanding those parts. What they DON'T get is the middle number.
The middle number, say it's 45, is NOT IN ANY UNIT OF LENGTH AT ALL! It's actually a ratio, in percent. It's a ratio between the width of the tire and the sidewall of the tire. So, for any tire with a middle number of, say, 45, the sidewall is 45% of the width (and there's a sidewall on the top and on the bottom when it's mounted so it basically contributes twice to the total tire height/diameter).
So, our 225/45R17 tire is 225mm wide, and each sidewall is 45% * 225mm, or 101mm. That's about a 4 inch sidewall (divide by 25.4) and a 8.9 inch width. Add sidewall + wheel + sidewall to get the total tire diameter, 4 + 17 + 4 = 25.0 inches.
Repeating the same for 235/45R17, the tire is 235mm wide, and each sidewall is 45% * 235 = 106mm. That's about a 4.2 inch sidewall (divide by 25.4) and a 9.3 inch width. Add sidewall + wheel + sidewall to get total tire diamter, 4.2 + 17 + 4.2 = 25.3 inches (removed rounding)
If you do the math, the 235/45R17 tire is 0.36 inches taller than the 225/45R17 tire, or about 1.4% taller.
On a more extreme example, all '45' sidewall tires are not the same. A 155/45R17 tire is 22.5" tall while a 305/45R17 tire is 27.8" tall. One has a tiny tiny sidewall, the other has a fat sidewall. The only thing the ratio is useful for know independent of the width is for determining wheel size. both the rediculous hypothetical tires I just made up would require a wheel width that's within the same range as the tire width. The tire has a range of about 5-30% wider than the wheel width for almost all '45' sidewall tires.