As Jason Nichols says, these terms are often used interchangeably.
The general theme is that pretty much anything can be called a "peg" (except perhaps the case where two countries are literally using the same currency), while "fixed" tends to refer to institutional arrangements that are more automatic, where changes in the exchange rate are perceived to be less likely. "Peg" tends to be used when some entity (e.g. a central bank) is doing the "pegging"; the more active and less automatic the behavior of this entity is, the more likely that you'll call the arrangement a "peg".
To further explain the (inconsistent and informal) differences in usage that do seem to exist, I have to describe several different kinds of exchange rate regimes.
- At one extreme, country A may use the same currency as country B. Relevant cases include the Euro Area and dollarized Western Hemisphere countries like Panama and Ecuador. In this case, we might say that A and B have a "fixed" exchange rate (at one to one), but we probably wouldn't say that their rate is "pegged". A similar but less certain case would be, for instance, the CFA Franc in Africa, which is in principle distinct from the Euro but has a fixed Euro conversion rate and is guaranteed by the government of a country using the Euro (France). This might be called "fixed" or "pegged".
- A less extreme situation is a currency board, where country A has a different currency than country B but promises to always convert them at a certain rate, and has reserves denominated in country B's currency backing up every unit of its own currency so that (in principle) this promise can always be fulfilled. Examples include Argentina's defunct currency board (a good example of a case where this promise was ultimately not fulfilled in practice) and Hong Kong's current currency board. I've seen both "fixed" and "pegged" used to describe such arrangements; as mentioned above, my sense is that "peg" is more common as a descriptor when the arrangement is perceived as being less automatic, with its permanence less certain.
- Still weaker is the situation where country A sets a certain exchange rate with country B but doesn't have a formal arrangement like a currency board to back it up. This is extremely common; one rare example among developed countries is Denmark, which pegs to the Euro. This is the last case among (1)-(3) that can still often be called a "fixed" rate or a "peg". The line between this and a currency board in (2) is often blurry, since in this case too central banks often maintain large foreign reserves, perhaps enough to back up every unit of their currency outstanding; but usually this case involves a less universal guarantee of convertibility.
- Then there is a vast set of arrangements where the rate is not perceived as being truly fixed for the indefinite future, but instead is allowed to fluctuate within a moderate or large band, or is subject to a "crawling" peg that is adjusted by the central bank, or is pegged to a possibly malleable basket of currencies, etc. These arrangements are usually called "pegs" rather than "fixed", because (after all) they aren't that fixed! A conspicuous modern example is China.
- There are even looser arrangements where a central bank floats the currency but pays some attention to exchange rates and wants to avoid fluctuations that are too large (rather than exclusively hewing to some domestic objective like an inflation target), using both the tools of domestic monetary policy (interest rates) and intervention in foreign exchange markets to keep exchange rates in line. This is usually described as a "managed" float, "dirty" float, or something similar; it would rarely be described as a peg, but sometimes the line between (4) and (5) can be blurry.
- Then at the extreme other end, we have countries that float their currencies and do not routinely intervene in foreign exchange markets. These arrangements would never be called either "fixed" or "pegged".
To sum up, my overall impression is that it's pretty vague, but that among 1 to 6 above, you're more likely to call something earlier in the list "fixed", something more in the middle of the list a "peg", and something later in the list a "float". But I am not aware of any formal, clearly specified distinction between terms as general as "fixed" and "peg". When exchange rates are classified, as in the IMF's annual report on the topic, this is usually done in a much more specific and descriptive way.