Why infinities but not infinitesimals? Why is the use of infinities generally accepted, but not the use of infinitesimals?
I know a little about non-standard analysis, but that seems to be the exception?  From what I've seen, usually infinitesimals are avoided elsewhere (and are replaced by the notion of a limit, which again is based on infinities) even when they are presented rigorously and clearly.
Is there a valid reason for this (beyond simple preference)?
 A: This is a tough question that's hard to answer definitively because of the different "infinities" (for an overview, see Understanding infinity), the history and popularity of different branches of math involved, etc.
I think a large portion of the reasons come down to the fact that there are more contexts in which "infinities" would be useful than "infinitesimals". This has knock-on effects for, say, how math curricula are designed in universities, the level of general awareness of mathematicians which affects their ability to spread ideas, etc.

Infinitesimals don't arise in common contexts
Ordinals were discovered when Cantor was working on real analysis, and cardinals (especially the countable-uncountable distinction) are often useful when dealing with infinite sets, both in and outside of analysis. And $\pm\infty$ in the extended reals help to give a tidy account of limits and measure. And if we broaden out view to complex analysis, the Riemann sphere is fundamental and has a point labeled $\infty$. But none of these contexts directly lend themselves to an infinitesimal.
For ordinals and cardinals, we don't even have something positive but less than $1$. And for the others, an infinitesimal would break the (Dedekind) completeness property of the reals that is critical for usual analysis to work.
and their use is limited
Now, you can change the arithmetic on the ordinals to get the surreal numbers, or look at other non-archimedean fields, perhaps in a more general/abstract way. But these are not often useful for analysis purposes. In Combinatorial Game Theory, there are infinitesimals like "up" that don't reside in a field, but that's a pretty niche area/application.
except maybe in nonstandard analysis
Arguably the most useful example of infinitesimals would be in Robinson's hyperreals for nonstandard analysis. In the scheme of things, this is relatively new in Calculus (so it's unfamiliar to many teachers and students would still have to learn standard approaches to connect with other material), and doesn't give you any new theorems about analysis, so it's tough to introduce into a curriculum. It's also arguably harder to make fully formal than a traditional construction of the reals.
That said, some mathematicians are using nonstandard analysis in their arguments. For example, Terry Tao has a number of blog posts about it.
