等しい and 同等: equal items and comparable levels
Read 等しい and 同等 within the stated comparison. Preserve ほぼ and 性能の面では so that approximate equality in one aspect does not become complete identity.
等しい with lengths and opportunities
等しい, read ひとしい, means equal or equivalent. 二つの線の長さはほぼ等しい。 says that the two lines are almost equal in length.
全員に等しい機会が与えられた。 says that everyone was given an equal opportunity. 等しい is not limited to numerical comparisons such as length.
同等 compares a specified aspect
同等, read どうとう, means equivalent or comparable. 二つの製品は性能の面ではほぼ同等だ。 says that the two products are almost equivalent in terms of performance.
性能の面では limits the comparison to performance, and ほぼ retains the qualification almost. Do not extend this to identical prices and dimensions.
Keep the scope and degree of comparison
The line sentence compares length; the product sentence compares performance. Neither word alone extends the comparison to every property of the objects.
Giving people equal opportunities also does not establish that everyone obtained the same result. Distinguish the item compared or provided from subsequent outcomes.
If two products are 性能の面ではほぼ同等, must their prices and dimensions be exactly identical?
Show the answer and why
You cannot conclude that. The original states only approximate equivalence in performance. Check prices and dimensions separately.