These two have the same meaning which is "conveniently" but I'm not sure if there is any difference in using these two.
How should 顺便 be used?
顺便 does not mean "conveniently" per se. One can say, 从我家去地铁站很方便, but one cannot say 从我家去地铁站很顺便.
Rather, as stated by Shudong's answer, 顺便 implies that the action is part of a larger set of events. Moreover, this action is something of an "afterthought", something we might as well do because it's convenient (as a result of the larger context). For instance:
Today I'm going to see my advisor, so I might as well stop by the university library to borrow a book.
These dictionary suggestions of "on the way" or "by the way" can thus also be more appropriate than "convenient", although they are rather lacking.
"on the way" suggests some sort of ordering that may be inappropriate (since that would imply that whatever is "on the way" comes before the more important event). (For instance, in my example, it's possible that I might go to the library after seeing my advisor; there's nothing to suggest otherwise.)
"by the way" doesn't give us the full picture. I would translate 顺便问一下 more closely as "while I'm at it, I might as well ask", even if "by the way" does work as a gloss of the entire phrase (once an appropriate verb -- here which should be related to speaking -- is added to 顺便).
What is the meaning of 随手?
My dictionary gives "at one's convenience" as the definition for 随手. This can work for an approximate gloss, although "carelessly" (or words with the same connotations) tend to fit more the scenario consistently. For instance, using examples taken from here:
废旧电池即使没有了使用价值也不可随手乱丢，因为它们会污染环境！Old batteries can't be discarded casually even when they are no longer useful, because they'll pollute the environment!
"at one's convenience" still works, because "at one's convenience" would imply a certain lack of care.
护林人随手扔下的一个烟头居然引燃了这场森林大火。A [cigarette butt] left in passing by a forest ranger could have caused the forest fire.
"at one's convenience" definitely doesn't work here. Nor does "conveniently".
However, "carelessly" still fails as a translation in cases like 请随手关门 ("please close the door on your way out"). There are similar examples in the source I linked to. Given all this, it would seen that there's no one-size-fits-all translation for 随手. However, we can at least deduce that it is used for actions that can be done with a lot of convenience, which is also why "carelessly" is sometimes a good fit.
Moreover, 随手 is restricted to actions that involve the hands. For instance, we cannot say 随手问. (As pointed out by NS.X., there is also 随口 or 顺口. As with 随手, 随口 tends to only be used for the relevant body part, here things related to speaking, although it can be more consistently interpreted as "careless" than 随手 can be. While 顺口 can mean something like "flowing" even if a similar lack of restraint can be implied in the right contexts, it also is restricted to things that might involve some sort of speech.)
So what's the difference?
随手 does not imply that the related action is connected to a larger set of things. The example sentences with 随手 would still make sense out of context if I removed the unrelated clauses, whereas a sentence with 顺便 must be at least be understood to fall within a larger framework. (I can say 我顺便去图书馆借本书, but it's understood that I am also going to do something else because of the use of 顺便.)
"conveniently" is a poor fit as a direct translation for both terms, for the reasons I described above. In terms of clearly connoting convenience, 顺便 does a slightly better job, but it can only be used in specific scenarios. Moreover, the convenience aspect when 顺便 is used may be a bit more dependent on context than when 随手 is used -- it's convenient for me to go to the library because I'm already meeting my advisor, whereas it's almost always convenient to simply throw batteries in the trash.
Finally, 顺便 can generally be used for any types of actions, whereas 随手 cannot be.
顺便 seems to imply a inclusive relations, e.g. 我顺便回家 'I 顺便 go home', in which 'go home' is located in a bigger event unuttered. 顺手, by contrast, relates to immediacy, e.g. 我顺手关门'I 顺手 close the door.' However, it also sounds OK if we speak 我顺便关门, and the sentence *我顺手回家 sounds very bad. This asymmetrical accepted replacement indicates that to some degree their usages overlap.