"저는 밥을 먹으면 행복해요."
Translation:I am happy when I eat rice.
It is different; ~면 is the conditional conjunctive, meaning that if the first verb is met as a condition, the next verb is true (if A, then B). ~면서 or ~며 means "at the same time" (while A, also B), which is just a matter of time and doesn't mean A and B are connected. ~면서 can also mean "though, yet, in spite of."
"When" and "if" are equal in this lesson's translation into English. This Korean sentence is about the chance or opportunity to eat rice because "If" is about the uncertainty of getting rice.
"If I eat rice, I am happy"
"When I eat rice, I am happy" (*)
On the the other hand, "while" and "when" are equal to "during". You certainly have the rice and definitely are in the act of eating rice.
"While I eat rice, I am happy."
"When I eat rice, I am happy." (*)
........... ( * The use of "when" is less exact than using "if" or "while" in English.).