I've found a complete answer to my question in this paper:
Ardourel, Vincent; Jebeile, Julie, On the presumed superiority of analytical solutions over numerical methods, Eur. J. Philos. Sci. 7, No. 2, 201-220 (2017). ZBL1384.00034.
There are many differential equations for which an analytical solution is known, but the Taylor series for computing the function converges so slowly that the solution is of no practical use.
Examples given by the authors are analytical solutions to the N-body problem proposed by Sundman (1907, 1909) and Wang (1991):
There are analytical solutions to the N-body problem (Sundman 1907, 1909; Wang 1991), which are perfectly general and have been viewed as theoreti- cal successes. However, these solutions remain useless for physicists: neither Sundman’s solution nor Wang’s solution can be used to compute the trajecto- ries of an N-body system. This failure is intrinsically due to the form of these solutions, which are infinite convergent series in powers of t1/3, and thus have no practical use to make quantitative predictions. As Wang himself writes:
Although the conclusion given here provides a way to integrate the N- body problem, one does not obtain a useful solution in series expansion. The reason for this is because the speed of convergence of the resulting solution is terribly slow. One has to sum, for example, an incredible number of terms, even for an approximate solution of first order in q, p, t. (Wang 1991, p. 87)8
As Florin Diacu asserts about Sundman’s solution and Wang’s solution:
One would have to sum up millions of terms to determine the motion of the particles for insignificantly short intervals of time. The round-off errors make these series unusable in numerical work. From the theoret- ical point of view, these solutions add nothing to what was previously known about the n-body problem. (1996, p. 70).
Another example given by the authors is the Airy function, for which we have an analytical solution whose power series converges analytically, even though it doesn't converge in a floating point calculation:
Infinite series are not always convenient for numerical application. The reason,
as Fillion and Bangu claim, is that one cannot sum an infinite number of terms.
In such a case, one can only sum the terms of a truncated version of the series.
But such finite summation may lead to an inaccurate numerical result. Fillion
and Bangu (2015) illustrate this with the Taylor series of the Airy function
(a special function). They write “[n]umerically, even if the series converges for
all x, it might be of little practical use, since the theoretical uniform conver-
gence might not translate to success in numerical contexts”(p. 6). Ai(−t) is
the analytical solution to d2x/dt2 = −tx. The equation describes an harmonic
oscillator whose stiffness increases over time. Fillion and Corless (2014) insist
that, although the infinite Taylor series of Airy function “converges uniformly,
the floating-point computation diverges” (p. 1461). This means that a numer-
ical computation of the Taylor series of the function – e.g. Ai(−12.82) – with
floating-point arithmetic does not necessary tend to the exact value while the numerical computation includes more and more terms of the series. Such a numerical computation thus seems to be useless for the purpose of making quantitative predictions.