In a few of my entries on Lang-8.com, several of my Chinese friends have changed 平常 into 通常 and vice versa without providing additional explanation. It it just a matter of personal taste?
The two words share the same character "常“(common, usual,eternal), so the meanings overlap somewhat. Both of them can mean "commonly, usually", however, they are not exactly the same.
"平常“ means (notice 平 means "plain，commonplace" here）
[adj]ordinary, casual/informal [adv]ordinarily,in a casual/informal way
[adj]common,usual [adv]in general, generally, in most cases
When something is ordinary, usually you could say it's the most possible case or it's common, that's how the two words have the similar meaning.
我平常坐公交车去上班。 Usually I go to work by bus ( in my ordinary life).
我通常坐公交车去上班。 Usually I go to work by bus ( in most cases).
通常情况下，伤口的血液会自动凝固。 Generally, the blood from the wound coagulates automatically.
平常情况下，伤口的血液会自动凝固。 Sounds weird to me.
真理通常掌握在少数人手中。 The truth are held in few people in general.
真理平常掌握在少数人手中。 Sounds weird to me.
我平常叫他老王，但是在公司里我通常叫他王经理。 I call him Lao Wang ordinarily(casually), but I would usually call him Manager Wang in the company(because this is a serious situation). Not interchangeable in this case.
我只想回到平常的生活。 I just want to return to a common\normal(in a style that average people have) life. Here, 通常 is not acceptable.
This is a very good question.
Both could mean "usually", "commonly", "generally", "expectedlly", "mostly"
Not only could the two phrases be used interchangeably in quite a lot of situations, it could also, like a chameleon, change from one to the other depending on the structure of the sentence itself.
(1) 男人先开口求婚是平常的， （It's common / usual for a man to ask for marriage)
(2) 求婚通常是男人先开口的, (marriage proposals are commonly / usually asked for by men)
In the English versions, they are virtually identical.
The differences in the Chinese versions are:-
In (1), it is to express the idea that in the usual, common course of events it would happen, or expected to happen; normally not arguable; exceptions are rarely expected.
In (2), it is to express the idea that most of the time it would happen, but arguable as exceptions are expected to or most likely occur.
So, 平常 = would occur or happen as commonly expected; exceptions are rare.
通常 = most likely to occur or happen with exceptions to be expected.
Admittedly not an easy question to answer. This is the best I could do.