Natural vs "Artificial"
All vaccines contain the same components as a "natural" infection, immunologically speaking: either a virus (killed or weakened); or a weakened bacterial toxin (toxoid); or the antigen-bearing part of the virus/bacterium (subunit); or a poor antigen attached to a carrier protein (conjugate)1,2. This is because of the way the immune system works and because the purpose of a vaccine is to teach the body's own immune system to detect threats and hence eliminate them before they can lead to an infection.
Antibodies and lymphocytes are made by the body itself, and they are no different in nature to what it would produce in the event of an infection a) because the stimulus is the same (even in an infection, the immune system responds to the antigen, not the entire organism--antigens are how the body distinguishes between self and non-self); and b) because the immune system is limited in the type of cells and antibodies it can produce. It can only produce those according to the blueprints in our DNA and they are specific to the antigen, not the vehicle/entire organism.
Comparison of Immune Response
As mentioned above, the cells are the same, the antigens are the same. That means a lymphocyte specific for a particular antigen will live for the same duration of time and function in the same way, regardless of the mode by which the immune system was exposed to the antigen. This is how we develop long-term immunity, whether by infection or by vaccination.
However, during an infection there are significant numbers of actively replicating/reproducing viruses or bacteria and/or a much larger quantity of toxins in the system whereas vaccines inject smaller amounts of weaker/dead organisms or antigen-bearing substances. This advantage is the body can eliminate the antigen more quickly, safely, and effectively than it can during an infection. The disadvantage, particularly in case of vaccines with killed viruses, is that the antigens are gone from the system sooner and so relatively fewer memory cells are made. (Vaccines using live viruses induce a higher, longer lasting immune response because the antigens persist longer3.) Remember, each memory cell lasts just as long as it is naturally supposed to--there may simply be somewhat fewer of them than made during an infection. This is perfectly acceptable since the purpose of a vaccine is not to provoke a greater response to itself but to keep the immune system prepared to mount a greater and more effective/co-ordinated response to the actual threat, should the system ever come across it. This is like the difference between war and war games.
Transfer of Maternal Antibodies
Whatever the stimulus for their production, the body only makes 5 different classes of antibodies; and only 2 of these can be passed on from mother to child. The first, IgG, crosses the placental barrier while the child is in utero. These antibodies are responsible for the passive immunity that provides infants some protection up to 6-9 months of age. The other antibody, IgA, is naturally found in body secretions in varying quantities. There is a small amount of it present in breast milk, but it cannot be absorbed by the gut because human infants do not have a porous stomach lining.
Impact of Maternal Antibodies on Measles
Regarding measles in particular, the only information I could find was it has been noted that vaccination before passive immunity from maternal antibodies wears off (~6-9 months) may lead to the mother's antibodies inactivating the vaccine before it has a chance to induce an immune response in the virus.4 There was no mention of a difference in this interaction depending on whether the mother's immunity was the result of a vaccination or an infection (keeping in mind, of course, that there is no difference in the antibodies but there may be a slight difference in memory cell/antibody levels). Other than that, there is no noted interaction between maternal antibodies and vaccinations or indeed the actual measles virus after passive immunity wanes.
The Immune System and Vaccination | Immunisation Advisory Centre
Understanding How Vaccines Work (PDF) | CDC
Vaccine Immunology (PDF) | WHO
Vaccine Types | Vaccines.gov