What's the difference between the Life Cycle Hypothesis and Permanent Income Hypothesis?

The ideas are very similar and functionally virtually equivalent although there arguably is subtle difference. For example, Romer in his Advanced Macroeconomics, which is widely used intermediate macroeconomics handbook, calls both the life cycle hypothesis and permanent-income hypothesis just permanent income hypothesis. To be more specific he says

Thus the individual uses saving and borrowing to smooth the path of consumption. This is the key idea of the permanent-income hypothesis of Modigliani and Brumberg (1954) and Friedman (1957).

Where the Modigliani and Brumberg (1954) refers to the paper where the life-cycle hypothesis originates.

There is very subtle difference though. Friedman's permanent income hypothesis, focuses more narrowly on income. Friedman's argument is also very 'instrumentalist' and it goes as follows. We know from macroeconomics that consumption depends on income so we will have:

$$C_t = a + bY_t + e_t$$

where the estimated $hat{b}$ is given as covariance of income and consumption over variance in income $hat{b} = sum(C_t-bar{C})(Y_t-bar{Y})/sum(Y_t-bar{Y})^2$ and $hat{a}= bar{C}- hat{beta}bar{Y}$. Now if we assume that $Y$ can be decomposed into permanent $Y^P$ and temporary $Y^T$ income (i.e. $Y=Y^P + Y^T$) and that the two are uncorrelated with transitory income having mean zero we can following Friemdman show that:

$$b= k frac{sum (Y_t^P- bar{Y_t}^P)^2}{sum ( Y_t- bar{Y_t})^2} text{ and } hat{a} = bar{C} -hat{b}(bar{Y}^P+bar{Y}^T)$$

Since transitory income is on average assumed to be zero, and because variation in transitory income which is component of the denominator in calculating $hat{b}$ is arguably much smaller than the variation in permanent part of the income one can conclude that consumption is determined virtually completely by changes in permanent income.

The life cycle hypothesis of Modigliani and Brumberg is more analytical and pays also attention to person's whealth. Following presentation in Romer's Advanced Macroeconomics, a person at any time $t$ a person would make expectation of his/hers lifetime income $sum^T_{t=1} E_1 [Y_t]$ together with any initial assets/wealth $A_0$ and their current consumption at the time $t$ will be just an equal fraction of all income (i.e. permanent income in Friedman's lingo) and assets for that particular time period. That is the consumption in present period 1 would be given by:

$$C_1 = frac{1}{T}left(A_0+ sum^T_{t=1} E_1 [Y_t]right) text{ and }$$.

This implies that people will try to smooth their income over their life as it implies that people will save when the income is higher than average income and dis-save when it is lower than average since:

$$S_t=Y_t-C_t= left( Y_t - frac{1}{T} sum^T_{t=1} E_1 [Y_t]right) - frac{1}{T}A_0$$

Thus both PIH and LCH are functionally equivalent and they are both consistent with each other. They both argue that consumption is determined by your permanent income/lifetime income including any inherited assets. The main difference is in presentation and arguably the mechanisms are subtly different. The Friedman's PIH presentation is more 'empirically oriented' and focuses on income itself (although arguably Friedman implicitely included any assets in $Y$). The LCH is more analytical and pays more attention person's wealth and later versions also to demographic factors. Since they are so similar as already mentioned before the ideas are usually treated simultaneously in textbooks and some authors such as Romer call both ideas permanent income hypothesis and I recall seeing the reverse as well.