It has nothing to do with the conflict with Borel which developed later, and one can find a pretty explicit answer in the aforementioned letters of Lebesgue to Borel.

(These letters were first [published][1] in 1991 in *Cahiers du séminaire d’histoire des mathématiques*; selected letters with updated commentaries were also published later by Bru and Dugac in an extremely interesting separate [book][2].)

In letter CL (May 30, 1910) Lebesgue clearly states:

> Poincaré m'ignore; ce que j'ai fait ne s'écrit pas en formules. 

(“Poincaré ignores me, because what I have have done is not written in formulas.”)

EDIT In interpreting this statement of Lebesgue I trust the authority of Bru and Dugac who in "Les lendemains de l'intégrale" accompany this passage with a footnote (missing in the 1991 publication) stating that 

> Dans [the 1908 ICM address] Poincaré ne semble pas considérer l'intégrale de Lebesgue comme faisant partie de "l'avenir des mathématiques", puisqu'il ne mentionne pas du tout la théorie des fonctions de variable réelle de Borel, Baire et Lebesgue.

I would rather interpret the meaning of "formulas" in the words of Lebesgue in a more straighforward and naive way. It seems to me that he was referring to the opposition which was more recently so vividly revoked by Arnold in the form of "mathematics as an experimental science" vs "destructive bourbakism".

By the way, it is interesting to mention that the first applications of the Lebesgue theory were - may be surprisingly - not to analysis, but to probability (and the departure point of Borel's [Remarques sur certaines questions de probabilité][3], 1905 is clearly and explicitly the first edition of Poincaré's "Calcul des probabilités"). Poincaré had taught probability for 10 years and remained active in this area (let me just mention "Le hasard" that appeared first in 1907 and then was included as a chapter in "Science et méthode", 1908 and the second revised edition of "Calcul des probabilités", 1912), and still he makes no mention of Lebesgue's theory. This issue has been addressed, and there are excellent articles by Pier ([Henri Poincaré croyait-il au calcul des probabilités?][4], 1996), Cartier ([Le Calcul des Probabilités de Poincaré][5], 2006, the [English version][6] is a bit more detailed) and Mazliak ([Poincaré et le hasard][7], 2012 or the [English version][8]). To sum them up, 

> [Poincaré's] seemingly limited taste for new mathematical techniques, in particular measure theory and Lebesgue’s integration, though they could have provided decisive tools to tackle numerous problems (Mazliak)

is explained by his approach of

> a physicist and not of a mathematican (Cartier).

 


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1110360
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=2406268
  [3]: http://www.numdam.org/article/BSMF_1905__33__123_1.pdf
  [4]: http://www.numdam.org/article/PHSC_1996__1_4_69_0.pdf
  [5]: http://preprints.ihes.fr/2006/M/M-06-47.pdf
  [6]: https://mathscinet.ams.org/mathscinet-getitem?mr=2647630
  [7]: http://www.bourbaphy.fr/mazliakfr.pdf
  [8]: https://mathscinet.ams.org/mathscinet-getitem?mr=3329414